How did the (Catholic) Irish go from escaping the potato blight and working as manual laborers, to electing one of their own to the New York City mayorship in 40 years?

I am currently exploring the rise to sociopolitical relevancy of the Irish Catholic immigrants that came to the United States during and after the potato blight of the 1840s.

As far as I can tell, it happened to be a case of "right place right time" with regards to their 'takeover' of Tammany Hall - where they, according to Glazer and Moynihan, represented 34% of the voting population in 1855. Moreover, Is this explanation all there is to it?

I also recall reading on many a social media post that Irish immigrants tended to take jobs in the police force and firefighter teams, since the work was often dangerous, risky, and low paying. Are there any academic sources that I could refer to that supports this claim?