The only problem I find with the pause button is that it interferes with the growth. If you pause it every single day, you'll get, like, two evolutions I think before you stay there because you pause it.
2 words. Pause Button.
Seriously, if your life is that busy pausing them would work. Somewhat. They probably wouldn't get hungry often enough to really have you taking care of them but if you only have an hour a day (you didn't mention how long you ave to play with them) you could unpause them (you didn't say really how many you have total) right before you have time to play with them, allowing them to get hungry.
Sorry if someone said this before me, I didn't read all the posts but, I too am busy. School, social life, etc. But before school I set the time to twelve midnight, and they sleep. They do wake up before I get home, but they're adults now so they take care of themselves for a while. If I'm going out to party or shop or whatever, and I KNOW I will be coming home after, I will fill up all the hearts and leave it at home. If I start to worry after prolonged periods of time, I can always come home early to check. Never came home early though.. Haha. If I'll be long for quite a long time, I'll do the same as before, but I'd take it with me. That way I can go somewhere to check it real quick and fix any problems. If I'm going on an overnight stay, I take extra caution. I'll check it every few hours, and keep it in my purse/pocket at all times! Of course, on mute, so not everyone hears it, but I always remember they're there and be sure to check them.
I'm SO busy, but my Tamagotchi's haven't died ONE SINGLE TIME while using those methods.