Yes, there is quite a lot of FM going on in this video.
Bells: yes! (Although a bit more Casio CZ-sounding, but that is practically the same thing) Sweeps at 0:24, sounds like a layered sound, but yes - sounds like some FM there!
The Analog Four does not have typical “FM synthesis” like the Yamaha DX/TX/etc series - their implementation is actually phase modulation due to the inherent issues with FM. Secondly, the LFOs in the Analog Four cannot operate at high enough speeds to achieve the sort of classic FM sounds you would expect from a TX81z or DX7.
However, you can get sort of close to some of the sounds if you program it very carefully. A bit too lengthy to go into detail about right here and now, but I think there are a few more ‘classic’ sounding FM sounds in the Mini FM soundpack I made:
Regarding the exact sounds used in this video, the A4 can make similar sounds for sure, but not exactly the same. Bell-tones can be achieved with AM (paired with oscillating peak filter tones), 80’s “brass” can be made like usual (subtractive), the long decaying FM sweeps can be achieved with a frequency locked LFO. However not straight forward to program. (expect for the brass I guess!)
And how does FM typically sound like?
Well, here is a good example of one of my favorite Yamaha FM synthesizers, TX81z:
And if you feel like learning more about FM, this is a pretty good video:
Is FM the best kind of synthesis ever conceptualized? Objectively, yes. 
Is the OPL3 chip in Soundblaster 16 the best FM-chip Yamaha has made to date? Yes.