I also thought the "Bagman" episode from Better Call Saul drew from this, and also from The Searchers. I believe Three Men and a Baby is considered a remake. While I like this version (there were several) of Three Godfathers a LOT and have recommended it more than once, especially as a Christmas film, an even better version is the 1936 one - darker, more painful, and thus earning its redemptive ending in a harder, more moving way.
https://www.imdb.com/title/tt0028367/?ref_=fn_tt_tt_2
It shows pretty regularly on TCM, keep an eye out for it.