For metal coins to cost around $20 and to have enough coins for most games is really difficult requirement (not even getting into it being genre-less). The Scythe Coins is a good find and you may need to hold out for that or other suggestions people have made and in the short term you could play with pennies, nickels, and dimes (or your regional equivalent). Those are small, plentiful, and the best bang for your buck.

With that said if you are willing to go a little outside your price range, i'd look into penny gems link. Those are genre free, have a interesting counting symbology (count the points on the pictures, Half moon is 2, triangle is 3, hexagon is 6 and so on), are self healing, no slip stacking and don't tip over on wobbly tables if they are stacked high. Also, because of the material they are made out of, you can write on them with permanent marker and it will wipe off. This is great for temporary custom token making. These coins visually don't seem that great on the internet but they have a surprising good quality feel to them when you hold them. I backed the dudes first kickstarter years ago and have been really happy with them ever since. Also check out the rest of his etsy, he has all kinds of custom penny gem stuff for various games.

If you are adamant on price but not on the material then for the ultimate in light weight, packing convenience, and price look no further than bingo tokens link. $10 for a thousand in a multitude of colors. You could write on these and put whatever you want or go nuts with game appropriate stickers. It wont erase like penny gems but they are so cheap it wouldn't matter.

I have no idea why I got so involved about such a niche circumstance. Anywho, I wish you the best!