Why do pilots normally fly by CAS rather than TAS?
The reason is somewhat deeper than just the indication. Up until Mach-related effects come into play, CAS is what the airplane 'feels' as speed, and consequently, is all the pilot needs to know. The TAS, or even better, ground speed, is needed primarily for navigation, but not for flying per se.

All the most important aerodynamic quantities (lift, drag, moments, etc.) are proportional to dynamic pressure:

$$q = \frac{\rho V^2}{2}$$

where V is TAS. So apart from TAS, they all depend on the ambient air density $\rho$. CAS is, essentially, TAS 'corrected' for the factor $\sqrt {\rho / 2}$, i.e. the effect of density. Instead of two parameters now you need one. But it doesn't mean CAS is calculated from TAS; rather the opposite: it is easier to naturally measure CAS (or q), and hence it is displayed (becoming IAS). To get TAS, you need to measure $\rho$, which is normally done via known pressure altitude (i.e. static pressure) and temperature.

If you climb at a constant CAS, your TAS will increase as the air becomes thinner ($\rho$ falls), but most importantly, the airplane handling qualities remain largely the same. But at some point, you'll hit the Mach limit and will have to obey it. Technically, the transition is governed by Mach itself rather than the altitude.