Funny that your first hunch is to blame "flaky" internet and not abysmal netcode 
The reason I couldn't connect is also really Nintendo, the game said you were using your nookphone and you can't connect then because who knows what.
As if me disconnecting should have any influence on your progress.
Its 100% Nintendoh
I mean I kinda assumed it's cause you're a continent away during a global bandwidth shortage. It could also be my internet. But I've been online about ~20 times and that was the only issues I've had except for the first day...
In any case, you're frustrated. I get that. I'm also frustrated. Sleep mode was even handled better in the previous 3DS game.
As a partial solution, I believe you should be able to leave someone's island at any time by pressing the Minus button and selecting it from the menu.