I’ve been upgrading several of my bikes (mostly older than your one) to 1x set-ups. The important thing is to consider what gearing range you need as you’ll still struggle to get the same range as a 3x system unless you go to 12-speed (when you can get a 10-51 cassette) but where you’d also need a new rear wheel I think. So far the best option I’ve had so far is 1×11 with a 34t chainring and a 11-46 cassette. That’s fine at the low end although a little short at the top end, although really not bad at all. That’s on a fairly XC type bike – it’s probably be just about spot on on a slower FS bike.

I’m just about to try a 1×11 set-up with an 11-51 cassette which I think should be pretty good with a 34t up-front.

We’ve got quite a few bikes in our household, from a fixed wheel pub bike to a 12 speed Trance 29er. One of them is a 3×8 Diamondback Apex Comp with full STX/RC.
It’s a lovely bike, rides as well as the day I bought it and the riding experience would not be improved one bit by spending a load of money on going 9,10,11, or 12 speed.
I know a front mech is a little bit more of a faff than 1-12 but last year I bought a new mech, cassette and chain for peanuts. Didn’t have to change the (gripshift!) shifter, get a new freehub, anything like that and it’s got a super wide range of ratios and super positive shifting.

The original post did say that the drivetrain was worn out though, so if changing it anyway then considering 1x is an option.

I’m not saying 1x is the only option (my Epic is staying 3×9 for the moment as going 1x isn’t worthwhile for it, as it’s not likely to be an upgrade) but worth considering.

I upgraded my Inbred to 1×10 (34t front, 11-42 rear) using mostly 2nd hand bits via Ebay, and my Soul to 1×11 (34t front, 11-46 rear) as well and neither was all that expensive.