TLDR: This post has spurred me to think about my own preconceptions regarding tuning and has me convinced now that moving your sight a bit for your broadheads probably isn't a bad thing if the broadheads are flying and grouping well. I think it comes down to how we often define "tuned".
I think the whole "if your bow is perfectly in tune, then it should shoot everything to the same point"....is probably misguided and depends upon how you define "tuned".....if you define "tuned" as "shooting field points and all broadheads to the same point" then it is true
Let's say you define "tuned" as sending a certain brand and weight of field point and your arrow as straight through the air as possible (let's call this State A).
Well, does that necessarily mean that you can strap any contraption on the front that has different dimensions, weight distribution, and more drag than a field point and the flight should not be changed from that perfect state with the field point?
Maybe not.
Perhaps with State A configuration you will send a big 2 blade a few inches low at 40 yards consistently. To "fix" this, you have to bring your bow out of State A and in a sense detune it to get the broadhead to plane upward a bit to counteract the other differences in the arrow. This state where field points and broadheads hit the same let's call State B.
Because bows that are in State A are sometimes either also in State B (with the right broadhead and vanes maybe) or close enough to State B that most archers can't tell doesn't mean that State A equals State B in all cases. I think we pick State B simply because of what we've been told on the internet and also because it is convenient to be able to switch heads back and forth, and it feels good to see no change (because it conforms to what we've been told as well).
Also, if you are in State B, then you'll feel like you are shooting broadheads well and when you switch back to field points you won't be in the best tune for fields points (State A) but field points don't plane, so you'll think you still have the perfect field point tune and won't notice much.
Lastly, half of the people on the internet that claim that their field points and broadheads shoot the same probably: 1. are shooting at close range or 2. can't shoot well enough to tell the difference.
If someone's standard group is a pie plat at 40 yards, then there is so much variation that they will have a difficult time picking up on a few inch drift.
Another caveat, I'd say what I wrote probably applies more to up and down deviation between field points and broadheads and also smaller deviations (I wouldn't buy into this line of thinking if there was like a 1 foot deviation).
Since OP can't get into State B, then this might indicate that his second setup is hypercritical to tune and maybe form. This would tend to make me shy away from it perhaps (I know I'm contradicting my previous statements here) because a bow and arrow setup that can shoot most things very close is probably less critical to form differences in the field.