The very way that rifles are designed - rigidity and tight tolerance through and through - means there is no such issue as a design choice for better consistency (unless we are talking rifles that intentionally sacrifice accuracy for weight savings). Most time when you are looking at an inaccurate rifle, the problem is quality control somewhere. In other words, for most accurate rifles, I do not think shooting in different position can inherently impact its consistency.
Having said that, I can only think of two things that can make you less consistent if you were shooting at less supported positions - Weight of the rifle and quality of the trigger.
First lets talk about weight. When you shoot with less support (for example, standing and kneeling without sling) the heavier the rifle, the more faster your muscles fatigue and the faster you run out of oxygen. They all result in shaky hands and unsteady trigger pull, which in turn makes your consistency suffer.
Now lets turn to trigger. For me personally, there are 2 attributes in a trigger that affects its shooting quality - cleanness of break and weight of pull. (I consider length of travel to be a matter of choice and does not inherently affect the shooting quality). When the rifle is shot from a more supported position (for example, when supported by a vice), even with a crappy trigger, your finger's trigger pull will move the rifle very little, and you can take all the time you need to carefully pull it pass the break.
When you shoot the rifle completely unsupported, your rifle is in constant but tiny movement. Shooting accurately is basically pulling the trigger at the precise moment of the cross-hair passing the target. The heavy pull combined with the uncertainty of when it would fire (as a result of mushy breaking) can easily throw off the timing or the rifle itself.
So, my conclusion is that if you want a consistent rifle, just pick a reasonably accurate rifle without too much weight, and put in a nice trigger. Luckily most 1000 dollar centre-fire rifle today are reasonably accurate and reasonably light. Further, Savage and Tikka have been heavily competing on making better factory triggers for about 10 years now. Most other brands of this price range have also improved their factory triggers.
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