OK, happy to help. I must say though that I'm no expert either - just a pilot who has spent some time dabbling in principles of flight. Since the likes of John Tullamarine and others of his calibre haven't commented yet, I'll keep going.
Bernoulli's equation assumes an incompressible flow, but as air is in fact compressible using the Bernoulli equation to deduce airspeed requires that compressibility be taken into account. ... Is that even close to an accurate description of compressibility error?
Yes.
Some minor corrections though (no pun intended...)
A pitot tube measures total pressure, not dynamic pressure. The freestream air has dynamic pressure and static pressure. As the air is brought to rest inside the pitot tube, Bernoulli tells us that 100% of the dynamic pressure is perfectly converted to static pressure, which the sensors then measure. This is also called total pressure or stagnation pressure or pitot pressure.
As you've suggested air is actually compressed in this process, which Bernoulli's description cannot account for. Thus the pitot tube measures a higher impact pressure than you would get by adding dynamic + static pressures.
The ASI over-reads at high TAS or high Mach Number.