I say Yes, and I'll give personal examples:

I used to shoot .006 arrows...and they were great with field points. However, with a fixed blade broadhead they were consistently inconsistent. One arrow would always hit for example 3" low and left, the next would always hit 2" high and right, and the next would always hit somewhere else. It made getting a few hunting arrows together that would actually hit in the same spot very, very frustrating and difficult. I used mechanicals as a "band-aid" for the problem.
Now I shoot .001 arrows...and now I have consistent broadhead groups from all my arrows. Life is better.

Spend the extra money and save yourself the headache.

Cheaper carbons are okay if do certain things. I'd recommend trimming the shaft to length by cutting from both ends. Spine testing and spine tuning is an absolute must.You'll have to throw away the ones that don't fall into a certain range. Shoot a slightly stiff spine with a very high FOC and helically fletched feathers. Do all this, and you can get some very nice groups with cheap carbons with broadheads. Don't do this, and suffer with eradict arrow flight when shooting fixed blade broadheads.