Some specific product recommendations are that I've been pretty impressed with the Microshift Advent 1x9 drivetrain I recently bought for next to nothing off a customer and threw on my hardtail that previously had 2x10 shimano gearing. In my case, I mostly appreciate that I now have a clutch, but it honestly just works fine. For 10 speed they make AdventX.

I recommend against spending money on a new drivetrain that wasn't really designed for 1x. This rules out basically any Sram 1x10 setup and most Shimano (except some of the most recent, like RD-M6000-GS, which is rated officially for up to a 42t large cog).

Generally the price of 11 and 12 speed 1x drivetrains has dropped a lot, making a lot of 1x10 setups not really that attractive by price (except the aforementioned Microshift options), but availability is weird. In particular, I'd avoid any 1x12 Sram setups in particular right now because parts availability is projected to be bad for quite some time yet.

If you want to get your existing drivetrain to work rather significantly better, you can probably get an inexpensive Shimano Acera or better derailleur or most any 8 speed Microshift derailleur and set it up and resolve your shifting issues. Shimano Tourney level derailleurs really are pretty garbage and shift unimpressively under the best of circumstances. Most front derailleurs are fine, but again, not that expensive if there's genuinely something wrong with it. If you want to try an intermediate step to see how you like 1x, get a new crankset with a narrow wide 1x chainring, and a slightly better rear derailleur set up correctly. Since you have a riveted crank, don't worry too much about getting a crank that will work with your existing bottom bracket length, although if you do, yahtzee. You can still stick with a relatively inexpensive square taper crankset. You can keep your FD in place as a chain guide to make up for the lack of a clutch, adjusted in place using the limit screws.