The reason for using air coils in LC tank circuits

The main reason why air core inductors are used in RF is that non-air core is nonlinear, it can saturate and it does not behave uniformly over wide bandwidth. I wouldn't make an erratic conclusion like : "I see there are almost anywhere air cores inside, which means low inductance, therefore we should put big caps to have the same resonant frequency."

All inductors are self resonant so choose an inductor that has a self resonant frequency sufficiently higher than the operating frequency of your circuit. Or, make use of the self resonance if appropriate to do so.

For a tank circuit in the collector or drain of a transistor, higher Q occurs when the inductor/capacitor ratio is bigger and this may be desirable in some applications. However, for a tank circuit in a quadrature detector (used in FM demodulation), higher Q occurs with a higher capacitance/inductance ratio and this is usually desirable for FM demodulation.

In other words, the target application dictates whether you want a higher L/C ratio or a higher C/L ratio.

There are really two things that you are trying to specify when designing an LC network, centre frequency and Q factor, which makes sense when you think about it because you have two unknowns, the inductance and the capacitance.

There are also typically non negligible secondary effects like self resonance and saturation that have to be considered, but in the first instance the required centre frequency and Q will set the required inductance and capacitance via trivial maths, and the technology will then be driven by secondary considerations (Size, mechanical limits, availability, self resonance and required unloaded Q).

You usually see air coils at real power, or VHF where the core losses from using a ferrite start to become painful, or where very high unloaded Q is desired.