... There's no difference in performance between RHCP or LHCP except that when flying around several other pilots who (usually) have RHCP
IF yours are LHCP you're less likely to receive (or transmit) any interference (to/from them) because of their signals being attenuated by your
polarization being opposite of theirs.
Omnidirectional "sticks or whips" (usually 2 or 5 dB depending on the make) suck ... period.
Even if a "mushroom" has only the same gain (about 3 dB is typical) being polarized (and generally more efficient) still helps to provides a better signal.
Most pilots use 2.4 GHz mushrooms for flight control and 5.8 GHz mushrooms for video TX/RX. Don't use the wrong frequency antenna for your application or you'll be worse off than if using a "stick".
Planar antennas (bottom pic) provide higher gain BUT they are NOT omnidirectional so they need to be (generally) pointed at the quad.
They will also pick up either "stick" or "any mushroom" transmissions regardless of polarization.
Helical antennas provide the highest gains of all but also have the most narrow field of reception for any antenna so they also need to be (rather precisely) pointed at the quad. They also need to have matching polarization to the antenna they're looking at.
Both planar and helical antennas are only ever used for receiving because of their limited "field of view" (to borrow from camera terms for sake of simplicity).
Pilots using diversity systems (basically two receivers both "looking at" the quad with a circuit that automatically switches to the better signal between the tw0 receivers) will typically have one planar (or helical) antenna plus one mushroom.
There's also another type of antenna, that (imho) surpasses the typical mushroom, which is called a "pagoda" antenna. The signal dispersion and reception
is much more uniform than mushrooms, which still have lobes, and they are also much harder to damage than mushrooms.
The two key points to note are are: 1.) Be sure to buy antennas with the proper connections. 2.) Match polarization between the TX and RX antennas.
Aside from that everything else is somewhat just a matter of choice and money .