I bet is has to still be faster then Windows loading and updating ...
how fast is the CPU and RAM size is what? just a basic Linux/GNU running fluxbox or windowmaker or i3 that does not take up too much to show on screen. as far as shut down - I use my power button that is my fastest shut down instead of using the menu select or a cli shutdown command.

Any drive will likely be only noticably slow if
a) there is a fault
b) there is a very large amount of data to transfer
c) buffering is not turned on.

This (5 years) is not that "old" a machine. Similarly to JeremyBoden - I've run linux on some 18 year old hardware (PII 400MHz, 192 MB RAM, 6 GB hdd ... - Puppy in frugal mode).

I've currently got some 2008-2010 era laptops happily humming on the latest versions of both Arch and Win10 ...

Must be some particular issue with your hard drives if it's really unbearably slow with Mint.

Lubuntu is a perfect distro for a 10 year old computer, ts lightweight and includes everything you need.
Works fine as a desktop pc, memory is more important then CPU.