So for a constant IAS, TAS increases in climbing is due to the reduction of air density, but nothing to do with drag, because drag is only proportional to IAS, is that right?
Yes, that's it. IAS is really a measure of Dynamic Pressure, which is ALSO what drag is proportional to, so drag remains constant for a given IAS.
So, if you assume constant power, you achieve a higher TAS as you climb at a constant IAS.
HOWEVER (With the possible exception of a turbo-normalized Piston Engine) available thrust REDUCES with altitude, so at some point you won't be ABLE to maintain a constant IAS, and your performance will begin to degrade.
Now, some high performance aircraft will reach their service ceiling (perhaps mandated because of Pressurization as an example) while still having enough power to achieve VMO- and thus will ALWAYS be able to achieve a higher performance by climbing.
Most, however, reach there "Best" altitude lower than their "Highest" altitude.