That's the only right way to do.... Faucets are not that expensive, if you think about it, you use it (almost) every day, if it holds 15-20years you're paying less than ten bucks per year for it... Even if you take a high quality expensive one.

Buy yourself a bit of peace and replace it with a decent, quality brand one which has a replacement parts supply ( some parts get bad faster than others but if it's already rusty, throw it out)

Edit: don't take fancy fashion colours, just plain chrome, the fashion faucets look nice, but if you want to replace one after a few years, you have almost certainly to replace the whole set of the bathroom, as those colours are out of production.

And don't take matte finishes, they draw calcination like a magnet iron shavings...