If the mouse was by HP, then chances are either Logitech or Mircosoft or some other big OEM (but really, I can only think of Logitech and MS as the only ones) are the manufacturers behind it and probably have similar models of their own.

I never got used to using side buttons. Two buttons and a wheel is all I need/like. However, I do dislike mice with horizontal side buttons quite a bit more than ones with vertical buttons. With vertical buttons, if the design is right, they just don't get in my way and I can still use the mouse normally (and usually ignore those buttons.) But with horizontal stuff, I keep mis-clicking them. Nothing like the badly-misplaced DPI buttons on some no-name "gamer" mice, though - those can be attrocious to use.

In regards to your right mouse button loosing contact...
Open the mouse and check it out. Does it use the regular rectangular buttons? If so, those are easy to replace... or even service! After a long life, you can open these and carefully wipe the contacts with 1000-grit sandpaper. Of course, only do that if that's even the problem. Sometimes the plastic on the top shell of the mouse can become worn/indented, preventing the mouse button from becoming fully pressed unless more pressure is put. For mice shell that use ABS plastic (vast majority of mice), this can be corrected by applying a few layers of PVC/ABS glue... or just glue a thin PVC or ABS sheet and sand it down.
FWIW, I'm still using my 2-button no-scrollwheel Logitech MS-34 ball mouse from 1999 as a daily driver. The left button has been opened a number of times now for cleaning (with 1000-grit sandpaper)... but it still works fine.

The first one I had with this design was Icon7 Z500. I don't know which company actually thought it up. Something went wrong with it, but I no longer recall what.

Actually, this current HP one, the issue was much worse at some point in the past, but it seems it's just me adjusting over time. I'm incredibly clumsy, afraid to open it to not screw up anything further. So I'd rather leave it alone until the need is not pressing.

I'm one of those people who use arrow keys for movement in games rather than WSAD keys, my hand just doesn't work there properly. And having 2 extra actions on the mouse is incredibly handy. Vertical movement of my thumb comes naturally while horizontal, not so much, something about moving that joint.

Yours is holding up well it seems, over quarter of the century.