so my question is, if going further south, can you somehow file for 6K
through the JFK VOR and then get higher? Every year I travel from
EWB to Florida, and have chosen the MANTA routing at 14K.

Here in the Northeast, if you don't file the TEC, you are probably going
to get it anyway, at least around the NYC area. For example, going from
Providence to BWI, I've got a choice of 6000' on V16, or if I want
higher I either get there via MANTA, which is some 60 miles offshore or
I get stuck with going north to Barnes (BAF) then V292 to SAGES to Lake
Henry (LHY) to get the long way around. If I want to go through NYC,
then my only choice is V16 at 6000'. Period. And that is regardless of
what I might file.