I wouldn't say physicists were 'happily' using renormalization back in the day. Rather, some of them were happy and many others were very concerned. Then, the ideas behind renormalization were formalized in terms of (poorly named) renormalization group flows, and by the early 90s it was on quite solid mathematical footing (though there remains to this day quite a bit of confusing folklore and fear around the topic, complete with plenty of bad pedagogy that has given many an undergrad as well as many the interested mathematician a lot of anxiety)
However, simultaneously with us coming to understand renormalization better, we've also come to realize it's really not such a big deal and it was supremely overused back in the day. Nowadays, most modern field theorists think in terms of 'effective field theories' and are not nearly so interested in trying to sum infinite perturbative series so we have a lot less use for renormalization (though it does still have its place)
That makes sense. I ran across the idea, informally, back in the 1990s in grad school. And what I heard then was concerning. I heard physicists saying that it was fine. But I also saw that their notion of fine and mine did not agree...
I had not really updated my understanding much since then.
However, simultaneously with us coming to understand renormalization better, we've also come to realize it's really not such a big deal and it was supremely overused back in the day. Nowadays, most modern field theorists think in terms of 'effective field theories' and are not nearly so interested in trying to sum infinite perturbative series so we have a lot less use for renormalization (though it does still have its place)