My thoughts:
I first learned machine work in a race engine shop in southern California. I worked for an NHRA Comp Eliminator team.
Indulge me for moment and this will start to make sense.
In any serious racing effort, no rock gets left unturned. Anyone with common sense will realize this pretty quickly. Now, not all those gains are huge or make a serious impact by themselves. It's not until it all comes together that the gain becomes significant.
Much is the same with guns. If you spent an infinite number of dollars on making your action the worlds most perfect in terms of fit and finish, but still have a crap barrel and a stock that fits only in the most academic sense, then it will shoot like crap.
Since your not really "competing" you have to decide the performance level/expectation for yourself. If I'm a pro stock car and I make 1200 hp at the flywheel well guess what? So does every other PS car. If I figure out a way to pick up a 13hp gain, then I really accomplished something because everyone else is still stuck at 1200.
Bench rest shooting is probably the closest comparision in this regard. Everything is neck and neck so small gains are more significant.
BUT, by itself a 13hp gain is pretty small.
A properly tuned up receiver is only going to compliment a good barrel. This can be looked at as the 13 extra hp. A good pillar bedding job, worth probably another 10hp. By themselves, they don't amount to much but with all this done and done right, you'll have a gun that used to print 1.25 minute of angle rifle (10 shots) drop down into maybe a half or even a 1/3rd moa gun. (again, 10 shots)
But now, just like your race car that needs high octane fuel, your engine (rifle) is going to demand premium ammunition components prepared carefully to realize the maximum performance.
Hopefully this analogy made sense.
The heart of a precision rifle's accuracy is in its barrel. So, if working with a budget, go barrel first.
~BUT (always a but) understand that you can't fit a barrel first, and then go back and have a full blueprint done on an action after wards. Not if the GS is going to clean up the threads on the receiver. You would either have to lop off the barrel tennon and rethread/chamber or replace the barrel.
The thread union between action and barrel is pretty important. By cutting on the receiver, your changing the minor/major and pitch diameter of the thread. The barrel won't thread on with near the surface contact that it should have and this can cause an accuracy issue (at best) and be a serious safety concern (at worst).
So, if you want it to be "really right" then you really need to do both at the same time.
Now, to get preachy. I realize we all operate in budgets. BUT (**** BUT again) If your budgeting your consumables (ammo components) in with your build then your already killing yourself. If you hot rod a car but can't afford the gas its going to require afterward, then it might be best to leave things as they are.
Not trying to be a jerk, just be realistic with yourself.