I often end up with some minute thickness variation along the length of the boards.
This is hand-tool woodworking, some variation from part to part is par for the course. We aren't machines after all1. So if the variations you're getting are already minute you're ahead of the game :-)
With regard to apron pieces, there's a bit of a dirty secret in that all four don't actually have to be the same thickness. This seems like an excuse for sloppy stock prep but sometimes it's necessary, e.g. to use the pieces you've prepped even if something unexpected happens like some tearout at the last stage of planing. Or, you have a limited supply of a given wood with the right figure, matching colour, or it's just a rare species. In either case, if one or more boards start out, or end up, thinner than the others then you just need to go with it.
What really matters with aprons is whether they appear to be even, which is down to them being flush to the legs, or set back an equal amount. So if you have some variation in thickness piece to piece you just mark out for the joinery on the legs to suit the thickness of each part and proceed accordingly.
Are there other hand-tools or techniques offering more fidelity?
There is a fairly standard modification2 that allows the pins in marking gauges to cut a finer and more refined line or you can use a cutting gauge instead (or a wheel gauge, which is a type of cutting gauge).
But even with regular pins the mark left by a marking gauge can be quite neat if you use good technique and as long as the pins are good and sharp.
The wood used does matter a lot here, it's actually really hard to get consistently clean lines in softwoods because the pin will easily sink into the soft earlywood but much less so into the harder latewood. And open-grained hardwoods like ash, oak or sapele are a bit more challenging to mark than close-grained woods like maple or beech with their more uniform texture and smoother surface.
I can correct this by planing many boards together in the vice after they're squared, or after a panel glue-up, but one drawback is that the final thickness is strictly smaller than the set thickness.
This is also fine, as again the intended final thickness isn't written in stone.
But if it bothers you having the pieces end up thinner — and it's fine to be a bit OCD about this sort of thing! — you might consider doing this procedure as part of your standard stock preparation rather than it being an additional step.
BTW for glued-up panels for many it is absolutely normal to do the final evening up/flattening/smoothing afterwards, as even using cauls or internal alignment aids you can't always guarantee every board will be dead flush with one or both of its neighbours — which is a must, even a 0.05mm / 1/500" discrepancy can be easily felt with the fingers even if not easily visible!
1 But because we're dealing in wood even if you did size parts by machine they can end up varying somewhat, because of differential wood movement.
Source: Using marking gauges on The Valley Woodworker blog.