Let's say there is a difference.
All things being equal, the fact that it's up to debate seems to point to their not being much difference.
Things like individual craftsmanship and body style have more of an effect on the over all harmonics and volume than other factors like materials. Even then the flute difference will be subjective, and what I hear will be different than that of an older person, and a kid will be able to hear something different, so even an objective measure like decibels may not hold up subjectively. With such variability, I'd say listen to what the maker says. They have the chance to compare apples to apples, and if they say they cannot make an unlined flute that plays the same as its counterpart, then it seems best to defer opinion to them. Even then their views are probably filled with expectation bias and other perception filters.
The clear options to me (for cutting through) seem to be, change embouchure, kick the tune up an octave, get a Böhm, get a piccolo, find a session with fewer instruments, or get an amp. From the few flutes I've blown across the color of the wood, the metal in the rings, and other material choices have been way overshadowed by the player and design parameters to the point that the other factors would drop out even when played solo.
To Quote Calculus Made Easy by Silvanus P. Thompson
An ox might worry about a fea of ordinary size--a small creature of the first order of smallness. But he would probably not trouble himself about a fea's fea; being of the second order of smallness, it would be negligible. Even a gross of feas' feas would not be of much account to the ox.