If this were done exceptionally well--a new building in the character of the area, no parking lot, no gas-- maybe.... But when is the last time you saw one done well in the state of Florida? When I drive by convenience stores in the area (Springfield, Park/Oak Street in Avondale or Riverside), they seem to be dedicated magnets for dealers, street walkers and late night booze hounds. No matter how nice the signage is, this will not bode well for the neighborhood. Again, we can do better. I hope RAP steps up.
Those buildings that most convenience stores are in were built long before there was design review in the district. The majority of those buildings wouldn't fly today, so it's kind of an apples and orages example IMO.
You mentioned RAP - at this point, RAP is not taking a stance, as we are currently soliciting feedback from the neighborhood. With that said, if this does come to fruition, I have no doubt that RAP will definitely ensure that the building is built in full compliance with the design regulations outlined by the historic district and the restrictions in the zoning overlay. In particular, it will be built up to the street, and will definitely have appropriate signage. Look at the one in Boston as an example.
You mentioned that you've seen very few built well in Florida. I would argue that this is a result of the community not making 7-Eleven build to design regulations. If you force an urban, historic friendly layout, you might be pleasantly surprised at what you get.