There are a couple of different things coming together but the big one is that while they were first considered degenerates, criminal, drunks, etc. they weren’t

There are a couple of different things coming together but the big one is that while they were first considered degenerates, criminal, drunks, etc. they weren’t denied all of the legal status whiteness offered. As states removed the property requirements, allowing all white men to vote, the large groups of Irish immigrants voted cohesively enough to become very influential in a lot of cities – helpfully around the same time that booming cities needed professional police & fire departments, creating a ton of civil servant jobs which did not require formal certifications or uncommon skills. Once a few people from a tightly-knit community get in more will follow, and the Irish tended to be more insular as Catholics in a Protestant country.