If you plan to just have the rear shifter working, it probably makes sense to get the front working too, since most fair-to-decent shifters come new as pairs. I'm trying to picture what's the bike setup and whether you kept your derailleurs or "removed them." I wonder how you maintained chain tension and whether you would simply cross chaining for all these years. If you did "remove" them, then again, I wonder how you removed the chain to remove the derailleurs. So please define "removed them"; what does that exactly mean? You removed the shifters and cables? So were you riding the small ring up front?

So again, I'm suspecting both FD and RD (front derailleur and rear derailleur) are still present. Run new shifters and cables, lube, adjust, snip, cap, and you should be good to go. Note however that the shifters must be compatible with the rear derailleur for indexing to work. Falcon brand shifters should be Shimano compatible. Some SRAM MRX grip shifters are compatible too, but not the non-MRX ones. They work only with SRAM RDs. So you need buy both compatible RD and shifter for it to work and it must match the number of gears in the rear.

Otherwise, like other suggested, Tourney TX-35 or I'd go with Altus. Then a new chain. The thumb shifters, but not integrated with brake levers. I'd get separate levers, just because again, here, we don't know your brakes and if they are short pull cantilevers or long pull V-brakes/Linear Pull brakes.