No, this wouldn't have worked; the charm protected the Dursleys in the same way it protected Harry.
For a start, magic in the Harry Potter universe just doesn't work that way. Spells have counters, weaknesses, but they don't have loopholes. Killing the Dursleys to get at Harry would violate the intent of the magic in the same way as, say, flying over McGonagall's chessboard, or destroying all the houses in Godric's Hollow instead of trying to figure out which one the Potters were hiding in, and with Harry Potter magic it is always the intent of a spell, not the literal wording, that counts.
But that's just guessing, and we don't need to guess.
These are Harry's own words, in Chapter 2 of Deathly Hallows (emphasis mine):
Harry pressed on remorselessly. 'Once I'm seventeen, the protective charm that keeps me safe will break, and that exposes you as well as me.'
If breaking the charm exposed the Dursleys to attack, it stands to reason that the charm was indeed protecting them as well as Harry up until that point.
To follow up on some of the discussion in the comments, note that in the quote in the question, Dumbledore talks about "the place you call home" which does not necessarily imply that the protection is limited to the inside of the house.
While you can still call home the place where your mother's blood dwells, there you cannot be touched or harmed by Voldemort.
If he'd said "the house you call home" then, yes, we'd indeed have to assume that the protection lapsed the instant you walked out the front door, and conclude that the Death Eaters should by all rights have captured or killed Harry long before he had his run-in with the Dementor in Order of the Phoenix. But the place you call home is a broader concept. It can mean a house, a neighbourhood, a town, sometimes even an entire country.
Of course, Dumbledore's choice of words doesn't prove that Harry and the Dursleys were protected while they were out of the house. But they're consistent with it, and I think this interpretation makes more sense, given that the Death Eaters weren't camped outside Harry's door all summer.