Short answer: Yes, but...
The first guitar I bought was a Peavey Predator, a bottom-of-the-line, budget Strat clone. Several years later I came across a good deal on Fender Custom Shop Strat pickups, so I got them to put in the Predator. Yes, it did improve the sound a lot, but like your guitar the frets were very worn, the tuning machines were terrible, and the sound still wasn't quite right. So, I bought a new neck online and new locking tuning machines. That improved the playability and feel a lot, although it didn't do much for the tone. But now I play that guitar a lot more often than I used to and I've used it on stage and in recordings before.
So yes, you can piecemeal upgrade a Strat copy and get incremental improvements as you go.
There are some caveats, though:
- Parts won't always fit just right. I had to play around with the pickup covers on the Fender Custom Shop pickups because they didn't quite fit in the Peavey pick guard slots and the Peavey pickup covers wouldn't fit over the Fender pickups. I also had to shape and trim a piece of wood to go between the neck and body because the Peavey neck slot is about 3/8" deeper than a Fender Strat neck slot.
- The tone is better, but not yet where I want it to be. Everything affects tone, so I would have to replace the bridge, pick guard, electronics, and body before I would really get a huge improvement in sound. That's the same as a whole new guitar.
- The neck I bought came with a pre-slotted nut (not all do) but the slots were cut very shallow (very high action) since the neck manufacturer has no idea what kind of action people want or need or what works for the guitar. I actually have played around with guitar building so I had some nut slot files and I was able to deepen the nut slots fairly well to make the action playable. Others may have to take the new neck to a pro to have the nut slots adjusted which adds to the cost of buying a neck.
- In your particular situation, if you buy a standard Strat clone neck, it won't help very much, if at all, with detuning when you use the vibrato tailpiece. Using locking tuners and getting very smooth string trees would help more, as would a graphite nut (which would probably have to be professionally shaped). Or you could get a neck with a Floyd-Rose style locking nut and that would make a huge difference in tuning stability, but you'd probably have to replace the bridge also, since locking nuts require tuning adjustment at the bridge.
- In the end, once you have replaced all of the parts of the guitar, it probably won't be quite as good as if you had saved up and bought a higher end guitar all at once, and it will cost about the same or even a little more (based on some math I've done over the years to see if it's really worth it). If that's a concern for you, I would suggest you make your own calculations, but I think you'll find it's not the best possible way to end up with a better overall guitar.