Ok, this checks out as expected (with respect to mdns)...
However, I would like you to have demonstrated a ping response using the Avahi hostname (foo.local) rather than by IP address.

And what command do I use.
Be as explicit as you like.

Well the printer Avahi hostname (along with other hosts on the network) should be discovered using something like...
You may need to install 'avahi-utils' package first.

Running the CUPS dnssd backend manually might also return useful info...

Once you have the printer hostname eg Brother_DCP-9055CDN.local, then you could do something like
*Don't copy/run this blindly though, you need to use the relevant Avahi hostname for your printer.

If you can ping successfully, that would be a good sign. Alternatively you could just manually configure the printer URI using 'socket://<Printer_IP_address>'

No, Ping is in no way related to cups or it's network foibles. Since the 2 IPs are on the same local network you are seeing what is probably the IP of your machine you are pinging from. Most networks are designed so that if a ping is unable to reach the destination the last reachable address in the route returns the unreachable reply because it does not know where to go. For example, if I ping my laptop from my workstation while the laptop is powered off the reply comes from my workstation IP.
You would need to check at the printer and see if the network is actually properly configured and if the printer can see other devices from that end. If your local network is using DHCP then the printer needs either a fixed IP or a reserved DHCP address to prevent an address change which can happen at some point.
My HP printer is capable of printing a network status sheet which has the assigned IP as well as testing network connectivity. I use a reserved DHCP address so it stays static.

As has been said, mDNS is known to be flakey at best, especially with dynamic DHCP addresses.

I agree that the best idea is to allocate a static IP address for the printer, or a DHCP reserved address (configured in gateway router). That's what most experienced admins would do. However, mDNS should work (firewalls allowing) too. I've used Avahi for years with no issues.

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Here is the answer I get to the command I entered
It's about an Epson M200 which I also have.
shakespeare@RYZEN:~$ sudo /usr/lib/cups/backend/dnssd
[sudo] password for shakespeare:
DEBUG: Querying "EPSON\032AL-M200DN\032\04079\05813\05849\041._pdl-datastream._tcp.local"...
DEBUG: Querying "EPSON\032AL-M200DN\032\04079\05813\05849\041._printer._tcp.local"...
DEBUG: sent=0, count=2
DEBUG2: query_callback(browser=0x5598de41e010, interfaceIndex=2, protocol=1, event=0, fullName="EPSON\032AL-M200DN\032\04079\05813\05849\041._pdl-datastream._tcp.local", rrclass=1, rrtype=16, rdata=0x5598de41f55c, rdlen=128, flags=5, context=0x5598de41d300)
DEBUG2: query_callback: "ty=EPSON AL-M200DN".
DEBUG2: query_callback: "product=(EPSON AL-M200DN)".
DEBUG2: query_callback: "usb_MFG=SEIKO EPSON CORPORATION".
DEBUG2: query_callback: "usb_MDL=EPSON AL-M200DN".
DEBUG2: query_callback: "note=".
DEBUG2: query_callback: "qtotal=1".
DEBUG2: query_callback: "priority=10".
DEBUG2: query_callback(browser=0x5598de41e010, interfaceIndex=-1, protocol=-1, event=2, fullName="EPSON\032AL-M200DN\032\04079\05813\05849\041._pdl-datastream._tcp.local", rrclass=1, rrtype=16, rdata=(nil), rdlen=0, flags=0, context=0x5598de41d300)
DEBUG2: query_callback(browser=0x5598de41d6e0, interfaceIndex=2, protocol=1, event=0, fullName="EPSON\032AL-M200DN\032\04079\05813\05849\041._printer._tcp.local", rrclass=1, rrtype=16, rdata=0x5598de41f918, rdlen=127, flags=5, context=0x5598de41d180)
DEBUG2: query_callback: "ty=EPSON AL-M200DN".
DEBUG2: query_callback: "product=(EPSON AL-M200DN)".
DEBUG2: query_callback: "usb_MFG=SEIKO EPSON CORPORATION".
DEBUG2: query_callback: "usb_MDL=EPSON AL-M200DN".
DEBUG2: query_callback: "qtotal=1".
DEBUG2: query_callback: "rp=printer".
DEBUG2: query_callback: "note=".
DEBUG2: query_callback(browser=0x5598de41d6e0, interfaceIndex=-1, protocol=-1, event=2, fullName="EPSON\032AL-M200DN\032\04079\05813\05849\041._printer._tcp.local", rrclass=1, rrtype=16, rdata=(nil), rdlen=0, flags=0, context=0x5598de41d180)
network dnssd://EPSON%20AL-M200DN%20(79%3A13%3A49)._pdl-datastream._tcp.local/ "SEIKO EPSON CORPORATION EPSON AL-M200DN" "EPSON AL-M200DN (79:13:49)" "MFG:SEIKO EPSON CORPORATION;MDL:EPSON AL-M200DN;" ""
DEBUG: sent=2, count=2
DEBUG: sent=2, count=0
shakespeare@RYZEN:~$

How do I get the thing to work?
Here is my /etc/hosts (with garbage at the front).
# The following lines are desirable for IPv6 capable hosts
127.0.0.1 1-edge-chat.facebook.com
127.0.0.1 2-edge-chat.facebook.com
127.0.0.1 3-edge-chat.facebook.com
127.0.0.1 4-edge-chat.facebook.com
127.0.0.1 5-edge-chat.facebook.com
127.0.0.1 6-edge-chat.facebook.com
127.0.0.1 7-edge-chat.facebook.com
127.0.0.1 8-edge-chat.facebook.com
127.0.0.1 9-edge-chat.facebook.com
::1 ip6-localhost ip6-loopback
fe00::0 ip6-localnet
ff00::0 ip6-mcastprefix
ff02::1 ip6-allnodes
ff02::2 ip6-allrouters
127.0.0.1 localhost kissmetrics.com

# Local router, DNS, DHCP
192.168.2.1 Router Router.local

# DHCP pool 192.168.2.96-127
# Nodes on LAN with fixed IP addresses
# Note:
# Netmask=192.168.2.0/25 restricts to 192.168.2.1-127 (LAN and internet)
# Netmask=192.168.2.0/27 restricts to 192.168.2.1-95 (internet, no LAN)
# Netmask=192.168.2.96/27 restricts to 192.168.2.96-127 (LAN, no internet)
192.168.2.96 WwwStation
192.168.2.97 M200 AL-M200DN
192.168.2.98 FileStation FS
192.168.2.99 DCP9055 DCP9055CDN
192.168.2.100 WebStation WS
192.168.2.101 HP7410 HP49BDF9
192.168.2.102 MediaStation MS
192.168.2.103 RYZEN
192.168.2.104 CRUZ
192.168.2.106 Tarja
192.168.2.107 NEXUS
192.168.2.110 Private2020
192.168.2.111 BraviaTV TV
192.168.2.112 PlayStation PS3
192.168.2.114 Pi1

The CUPS dnssd backend only picked up your Epson printer. The 'avahi-browse -art' command should enumerate all Avahi-capable hosts. Or just consider manually configuring as I suggested before.

A problem arose during the command earlier run, it has to do with the net address of the host.
eno1: flags=4163<UP,BROADCAST,RUNNING,MULTICAST> mtu 1500
inet 192.168.2.109 netmask 255.255.255.0 broadcast 192.168.2.255
inet6 fe80::c8ad:64f0:cdcf:e0da prefixlen 64 scopeid 0x20<link>
ether 00:d8:61:d8:fe:34 txqueuelen 1000 (Ethernet)
RX packets 790479 bytes 677902643 (677.9 MB)
RX errors 0 dropped 0 overruns 0 frame 0
TX packets 345532 bytes 53445387 (53.4 MB)
TX errors 0 dropped 0 overruns 0 carrier 0 collisions 0
device interrupt 16 memory 0xa3200000-a3220000

Here's the output from 'avahi-browse -art'

+ eno1 IPv6 RYZEN Microsoft Windows Network local
+ eno1 IPv4 RYZEN Microsoft Windows Network local
+ lo IPv4 RYZEN Microsoft Windows Network local
= eno1 IPv6 RYZEN Microsoft Windows Network local
hostname = [RYZEN.local]
address = [fe80::c8ad:64f0:cdcf:e0da]
port = [445]
txt = []
= eno1 IPv4 RYZEN Microsoft Windows Network local
hostname = [RYZEN.local]
address = [192.168.2.109]
port = [445]
txt = []
= lo IPv4 RYZEN Microsoft Windows Network local
hostname = [localhost]
address = [127.0.0.1]
port = [445]
txt = []
+ eno1 IPv6 Private2020 Web Site local
+ eno1 IPv4 Private2020 Web Site local
= eno1 IPv6 Private2020 Web Site local
hostname = [Private2020.local]
address = [fe80::211:32ff:fec8:9a4b]
port = [5000]
txt = ["mac_address=00:11:32:c8:9a:4b|00:11:32:c8:9a:4c|00:11:32:c8:9a:4d|00:11:32:c8:9a:4e" "secure_admin_port=5001" "admin_port=5000" "version_build=25426" "version_minor=2" "version_major=6" "serial=2020R5RGP1QY2" "model=DS1819+" "vendor=Synology"]
= eno1 IPv4 Private2020 Web Site local
hostname = [Private2020.local]
address = [192.168.2.110]
port = [5000]
txt = ["mac_address=00:11:32:c8:9a:4b|00:11:32:c8:9a:4c|00:11:32:c8:9a:4d|00:11:32:c8:9a:4e" "secure_admin_port=5001" "admin_port=5000" "version_build=25426" "version_minor=2" "version_major=6" "serial=2020R5RGP1QY2" "model=DS1819+" "vendor=Synology"]

I see no evidence of the Brother printer reported. However, I note that you have an entry for your Brother printer added to /etc/hosts
So why not just manually configure the CUPS URI?

It doesn't work.
I have tried and failed to do it.

I have two other laptops and they work adequately enough.

I'm not sure what you have tried and failed with here. Use the CUPS web interface to change the URI to 'socket://192.168.2.99' or 'lpd://192.168.2.99/BINARY_P1' (of course making sure that the printer really can be reached at that IP address first).

I have tried the CUPS web interface
'socket://192.168.2.99' and 'lpd://192.168.2.99/BINARY_P1'
neither one will work as both give the same result.
"Unauthorized"
"Enter your username and password or the root username and password to access this page. If you are using Kerberos authentication, make sure you have a valid Kerberos ticket."
The web page doesn't tell me any more than that (I'm not using Kerberos).

I've also tried to use lpoptions
shakespeare@RYZEN:~$ lpoptions -p socket://192.168.2.99
lpoptions: Unable to add printer or instance: No such file or directory
shakespeare@RYZEN:~$ lpoptions -p lpd://192.168.2.99/BINARY_P1
lpoptions: Unable to add printer or instance: No such file or directory
shakespeare@RYZEN:~$

I have a username and password.