It is the closest station, however CTDOT decided against a shuttle from the Windsor Locks station and instead decided that shuttle service between Hartford station, the Hartford convention center, and Bradley airport was a better option - the extreme limitations of the present Windsor Locks station is one of the reasons for the decision if I recall correctly (although there are others as well - namely an additional desire was having a convention center shuttle.) This decision was criticized by many although CTDOT has defended it saying their research stated that a Hartford shuttle was the way to go. Supposedly they are going to reconsider the matter once a new Windsor Locks station is built - if they did it would be a second shuttle and not affect the current service.
I imagine they charge for that shuttle to Hartford. If they ran one to Windsor Locks, it is short enough to be a complimentary ride. Maybe.
Do they also run a shuttle to Springfield from Bradley? I haven't been to Bradley in about 20 years...and that was on a diverted flight, that remained on the ramp, until it was cleared to JFK...
I do not know of a Shuttle from Springfield to Bradley unless the local Springfield bus authority, PVTA, operates one. It's also possible Peter Pan Bus Lines might run something but I do not know of anything. The Hartford Shuttle which I think they call the Bradley Flyer is the only one I know of.