What's the upside of L, S and C bands for military search radars?

Factual Questions
1

I understand why tracking radars would use X-band and higher the importance of resolution for tracking.

For a given antenna size, though, a higher frequency also means that the antenna gain is greater which means more range.

Most airborne platforms only have enough room for a small antenna so they use X-band for search, tracking and target illumination. They typically increase the time-on-target, frequency and PRF when they switch to tracking and increase frequency while using continuous wave for target illumination.

Most platforms large enough to carry a larger antenna use L, S or C band for search. For example:

Land-based platforms:

“An electromechanically scanned parabolic E-band (10kW power) target acquisition radar is mounted on the rear top of the turret that when combined with the turret front mounted J-band (150kW power) monopulse tracking radar forms the 1RL144 (NATO:Hot Shot) radar system”

“The upgraded vehicle is equipped with the ASADS Ka-band target tracking radar and the PAGE I-band surveillance radar”

Sea-based:
The Arleigh Burke class destroyers mainly use the S-band AN/SPY-1 and the C-band AN/SPS-67 for search ( AN/SPY-1 - Wikipedia and AN/SPS-67 - Wikipedia)

The Ticonderoga class cruiser uses the L-band AN/SPS-49 (AN/SPS-49 - Wikipedia)
So, why do search radars tend to use the L, S and C bands? They could use X band as well and get even more antenna gain with their bigger antenna.

From what I’ve seen, the attenuation in X-band isn’t that bad. Have I got that wrong?

Is it chiefly a way to defeat the low observability characteristics which are integrated into some fighters and bombers? Is, say, C-band significantly better at detecting low observability planes than X-band?

Is it because a larger beamwidth in a search radar is a desirable characteristic since it allows faster (if less precise) scanning of a given area?

I’ve read that a lower frequency will produce a stronger signal for a given power input. Is this true? If so, I have a difficult time figuring out the tradeoffs between the stronger signal of the lower frequency and the greater antenna gain of the higher frequency.

2

L S and C bands aren’t attenuated as much by water vapor which makes them better suited for long range search radar. X band attenuates more through the atmosphere, but is good enough for airborne radar. You are correct that the choice of using X band for airborne radar is based largely on the antenna size. X band is attenuated more, but it’s good enough for short to medium range airborne use.

3

I guess atmospheric attenuation becomes less of an issue as you go up in altitude also. A plane at altitude would only have to deal with sparse atmosphere when detecting another plane. When detecting a ship, the ship’s greater RCS would probably help the plane more than the ship’s lower frequency would help the ship.