I think it's clear the speed problem is in the warp core, because once they added the dead alien fuel the ship sped up with no issue.
I'd further say it's always
about the warp core. Or the structural integrity field. Or the navigation software, or whatever. The warp coils are demonstrably not something that the engineers would tailor for a specific performance range, as so many plotlines feature something going wrong and the ship suddenly flying off at much higher than recommended or designed speed.
In the real world, when something goes wrong with your engines, you don't start moving faster. You typically start moving more slowly, or stop moving, or very suddenly move in multiple directions simultaneously. But warp coils don't appear to be "engines" per se - more like propellers that never cavitate, or wheels with bearings that never seize, or sails that never tear.
As far as we can tell, if you cast your coil set properly, it's good for all speeds from zero to infinite, just like a calculator is good for displaying all numbers from zero to infinite, even if some other components might fall short of producing infinities or even particularly large finite outputs.
Probably it isn't advantageous, perhaps not even possible, to create low performance coils for low performance starships. Quite possibly (and even statistically evidently), any starship is warp ten waiting to happen, by the very nature of what warp coils are. That wouldn't be much of a dramatic cheat.