How to design a cheap sine-wave generator up to 200 MHz?
Your second idea of using switched low pass filters to pass the fundamental of a square wave is the way it is done in many commercial RF signal generators. It does depend how clean you want your sinewave to be. It's quite difficult using an economical version of this technique to get better than 40dB typical, 30dB guaranteed suppression of harmonics, but that sort of level is adequate for many use cases.

There are several tricks you can employ to reduce cost and simplify design.

The first is to use filters at half octave steps, at least for the higher frequencies. Although a square wave has nominally no even harmonics, this breaks down for practical devices with asymmetries and breakthough resulting in a significant 2nd harmonic. At some suitably low frequency you can go to octave bands.

The next is to use low order elliptic designed filters, which improve the steepness of the transition band, at the expense of 'come-back' at higher frequencies.

The next is to arrange the casacade so that the highest frequency (so the one at which you are likely to have lowest power and lowest gain) goes through the shortest, least lossy, path, and that you add further sections as the frequency drops. A fixed well-designed 256MHz 'roofing' filter at the start of the cascade will deal with the come-back of the 192MHz filter, those two handle the 128MHz filter, and so on down.

The next is to switch the filters by passing current through PIN diodes, which is cheaper and easier than using other switching technologies. Bias current passes through the filter series inductors so biassing at a specific point in the filter cascade switches on the correct part of the filter, and the rest off.

The last is to only take the filters down to some reasonable frequency, and do the bottom frequency range in one go with a DDS and a single low pass filter.