It is predominantly a MT gym in terms of striking. Personally, I started training in KB (at a TKD based school) then later transitioned to MT. I see the flicky kicks you are talking about. I think it is mostly a product of trying to go light as I don't kick pads that way. It is a good observation though, and I should be drilling proper technique even in light sparring.
Regarding closing the distance, I often find myself frustrated by opponents who move back a lot. This seems to come down to individual style. Some of my sparring partners are more likely to stand and trade or constantly push forward themselves, which allows me to work counters, slips, combos etc. But I have real trouble with someone who constantly moves backward. Feints don't seem to work - as soon as I initiate any type of movement, the opponent leaps backward and/or leans way back at the hip and throws a kick. It seems to be a very effective type of defence that I have trouble overcoming. Is rushing forward behind a flurry of strikes the best approach? I try this sometimes with varying success. Against a skilled opponent I worry I may be being baited into letting my guard down during a forward push and "rushing right into the trap" so to speak. Mentally I think I am still learning to gauge when the situation calls for cautious, judicious striking vs a more risky aggressive approach.
Gym is in Canada, they train BJJ too which is the reason for the flag.