That would depend on which play...and which members of the audience. But no, it's not just a latter day interpretation; it's part of the reason Oliver Cromwell had all the theatres in England closed when he came to power. Edward II by Christopher Marlowe is quite explicit in it's theme of homosexuality. The use of the name Ganymede in As You Like It is a classical reference to homosexuality--audience members would have picked up on that. There are also plays by Shakespeare's contemporaries that go further--like Galathea by John Lyly--in which two girls(played by boys) disguise themselves as boys to avoid being part of a sacrifice to Neptune, and while disguised, fall in love with each other. When they find that they are both girls, they are still in love with each other--in the resolution, Venus says that she is going to turn one of them into a guy(but keep in mind too, that all plays had to pass through censorship at this time; quick fixes at the end don't change the fact that the playwright chose to deal with these themes in the first place).
IJames I came to the throne in 1603 in England; he himself, though he had children with Queen Anne of Denmark, had a number of male lovers; he also loved theatre, and was the sponser of Shakespeare's company, which was called "The King's Men" during his reign. (It had been called "The Lord Chamberlain's Men" under Elizabeth).