Interesting thought. And makes some sense, in that as it currently stands, taking someone unawares is no different than just reacting fast... when it seems like there should be more benefit than that. You go through all the effort of sneaking up on a guard for a surprise attack, and all you get out of it is a higher initiative (which there was a good chance you were going to get anyway even without the surprise.)
I think my feeling is that I would give Advantage to those who have Surprise, rather than Disadvantage to those who don't. I also think that for ease-of-use... I would couple it to giving +20 initiative to those with Surprise, rather than the current -20 to those without. This way... what ends up happening is that everyone rolls initiative at the start of combat, and those who got Surprise add the 20 to their roll, and gain Advantage for that first attack of the combat.
It is easiest to remember to give one side both things (+20 and Advantage) than it is to give things to both sides (-20 to those surprised and Advantage to those who have surprise).