A couple things to note. Deficits are fine, the new system isn't about having no deficits at all. You get reductions to unhappiness from deficits from buildings, happiness buildings, and also on a city level a bit of deficit really doesn't matter too much, especially in a fully built city. You're also supposed to do the Public Works projects (that's why it is there), and the AI will do those constantly, do the 25% conversions, and even float a decent bit of unhappiness. FWIW, in my game literally every AI sat very close to 60% from mid-late game till the end.
Anything above 60% has no empire-wide effect, just slight reductions to production at the city level, which doesn't really matter much.
That said, if you're playing a warmonger I highly suggest having excess happiness, and keeping everything except your best couple of cities at capped pop growth to be able to float unhappiness from puppets, WW, and crappy new cities. In general, I suggest you start to cap cities when they can't keep up with culture/science needs. You could easily float that 3 poverty unhappiness and 3 distress unhappiness (after Military Academy's -2 distress) if you didn't have all that other unhappiness.