I know this isn't massively helpful, but unless other evidence can be found to identify the John Streate on the charter, it is impossible to say. Looking at the number of John Streets(Streat/Streete) kicking around in the 17th century I would say probably not.
I would also say this John Street is probably not the father of Anthony who went to VA either and has been linked to him due to his Gunpowder Plot fame.
I haven't looked at the source, which is apparently 'The Street Genealogy', but I would look closely to see what evidence they present for this family link (and evidence of use of arms etc); as a son not being mentioned in a will because he is '25 and thinking about going to VA, so not around' (see profile) is hugely fishy. And there doesn't seem to be a baptism etc. or anything else for Anthony the son.