ATC uses multiple kinds of antennas depending on the use case. The antenna that makes sense for a tower might not make sense for a TRACON communicating with a specific sector that is not directly over the antenna.
Note that "omnidirectional" is a bit of a misnomer when it comes to aviation. An "omnidirectional" antenna radiates equally in azimuth (i.e. every direction on the ground), but radiates less in the vertical direction than horizontally. While this is perfect for ground communications that don't want to waste energy transmitting into the air, it's obviously less ideal for ground-to-air communications.
Analog transmissions don't strictly have a bit rate. They do have a bandwidth (of roughly 8 kHz), which very roughly corresponds to somewhere in the ballpark of 100 kbps.
The power is laughably low compared to many transmissions. A few tens of watts is plenty since the transmissions just need to go through empty air. By comparison, there are FM radio stations (which are VHF transmissions just like airband communications) that transmit as high as 320,000 watts effective radiated power!
Tower transmissions can go up pretty high. The transition between tower and other controllers is done by instructing the pilots to change to a different frequency, not by trying to make the transmission ranges exactly right.