After about 5 years of trusty service, my computer is getting slow.

I'm suspecting the SSD might be in need of a wipe, and I'm not sure whether these things have a password in their SSDs, however it had a security lock (set by the bios, mind). So the security lockout is easily overridden by a sleep/wake-up cycle.

Now, I'm well aware that if I wipe my SSD, I need to do it using the firmware of the SSD, for otherwise I'll be wrecking havoc on it. Is hdparm the right way to go?

And also, if there's a password on its SSD, does anyone know how to reset it or know the password?

You'd need to expand on line one of your post - a few specs, a few symptoms. Meltdown & Spectre firmware updates slowed some cpus down significantly. About the last thing I'd suspect is the SSD which is probably newer than your box.

I've a puny enough laptop by standards - i3 twin core 2.4Ghz, intel HD4000 video which is 6 years old, with an ssd. Keep it low cholesterol. Gnome or kde slow things to a crawl. Virtualbox with windoze 10 likewise, seamonkey, and heavy stuff like that. Avoid high cholesterol stuff, don't buy a 4K monitor for a sucky video card, and you should be grand. Are you using high cholesterol software?

Well, the machine is from 2016, An Acer Aspire XC-780: an i5 with 8GB of RAM. SSD was in it out of the box (92GB SSD + 1TB HDD).

All the parts in the box are Intel.

System generally runs fine, but launch is fairly slow (boot to CLI takes 7 seconds (yes, by that time I'm already distracted ) from grub).

First thing after you start, run top in a terminal. Examine what's using CPU. Have you built a live usb system and seen how it behaves there? In theory each thread can run up to 100%. I have a twin core 4 thread i3, and I've seen 390%+ on the odd program.

A possible explanation for sudden slowness is malware. You could be mining bitcoin for some <expletive>. That sort of thing is best handled by a complete format & reinstall if it can't be done surgically.

I use Bleachbit to keep the fat down (just don't take any options that you get a warning).
I also keep the swappiness very low for high RAM machine like yours.