"Nothing technical stops CB from being FM (in many places it is), and SSB is technically just another (improved) form of AM."
No, FM is not allowed, see below.
SSB is basically AM with a suppressed carrier. (i.e. In SSB, Frequency deviation is the demodulated frequency, whereas with FM frequency deviation is the demodulated amplitude.) (You're not wrong in calling it improved, I'm just expanding on how for anyone else reading.)
FWIW, You're not allowed to broadcast music on CB, but that doesn't stop people either.
§ 95.971 CBRS emission types.
Each CBRS transmitter type must be designed such that its capabilities are in compliance with the emission type rules in this section.
(a) Permitted emission types. CBRS transmitter types may transmit only AM voice emission type A3E and SSB voice emission types J3E, R3E, or H3E.
(b) SSB requirements. Each CBRS transmitter type that transmits emission type J3E, R3E, or H3E must be capable of transmitting only the upper sideband with suppressed, reduced or full carrier, respectively, but may additionally be capable of transmitting only the lower sideband, with suppressed, reduced or full carrier, respectively.
A3E or A3EG -Ordinary amplitude modulation used for low frequency and medium frequency AM broadcasting
J3E - SSB speech communication, used on HF bands by marine, aeronautical and amateur users
R3E - SSB with reduced carrier (AME) speech communication, primarily used on HF bands by the military (a.k.a. compatible sideband)
H3E - AM, single sideband, full carrier
But yes, even narrow FM would be too wide at 10-15kHz.
§ 95.973 CBRS authorized bandwidth.
Each CBRS transmitter type must be designed such that the occupied bandwidth does not exceed the authorized bandwidth for the emission type under test.
(a) AM. The authorized bandwidth for emission type A3E is 8 kHz.
(b) SSB. The authorized bandwidth for emission types J3E, R3E, and H3E is 4 kHz.
"Enter the UHF version of CB which allows for FM modulation." That could be interesting, but would be even more line-of-sight than CB is.