If you're not offended by the concept like some people seem to be, there are advantages. As mentioned in a previous post, sometimes you don't have much or any time to play the game on a certain day. Sometimes you might also just want to take a break from the game for an extended period. With time travel, if you felt so inclined, you could time travel back to the last in-game day you played and ensure that you don't have any wilted flowers, weeds, or any unexpected move-outs. You might also enjoy unlocking stores, store expansions, and in-game events faster and not having to wait as long to get Public Works Projects recommendations if you want to add those to your town.
It's ultimately up to you, your personal life and how much time and interest you can dedicate to the game, how you play the game, and what about the game is enjoyable to you. If nothing else, you could try it for a few days and if you don't end up liking it, just time travel back to the proper day and continue on as you would. If you live somewhere that recognizes it, Daylight Savings Time is coming up on Sunday so it's as good a time as any to give it a shot.
The game is designed to be played in real time using your console's time setting. It is an inherent feature. If you decide to time travel, you would be playing the game in a way it's not intended to be played.
It's your game, your time and a minority of people on this forum will try to enable you, "it feels good", but I wouldn't.
False. It can be played either way, and it is as of now intended as such, because if you time travel enough villagers will comment on that being a rumor about you, like how when you're starting out they say you're a freshman mayor. Furthermore, if they absolutely wanted you to be stuck with the actual date and time, they wouldn't allow you to choose a different date and time in game, separate from the console's internal clock.