Hard drive removal

What kind of recycler? There are disposal services that physically destroy/shred hard drives. Destruction is the only effective way to prevent recovery/access to data. You can disassemble it and destroy the platters yourself. Search for "destroy hard drive" and other such terms.

If you're removing the hard drive, I don't think you need to do anything more.

Correct, there's nothing needs to be done to the computer itself after removing the hard drive(s). My reply above was a mis-reading that the concern is how to dispose of the hard drive (which may be a concern at some point).

There is free software available of a type that I think are generically called "wipe" programs. They perform numerously iterative rewrites of nonsensical data on each segment of the drive so that when done, no info present at the start is there or can be read.

Depending on the capacity and age, you might want to consider putting it in an enclosure and use it for backup. It can be disconnected after backups and be safe from malware/ransomware.