Especially if Mr. Angelino was pretty much likely going to fire Jack every time anyway?

I think Jack Tripper was very accident prone, hence his last name. Who else was willing to hire a klutzy chef?

And Mr. Angelino is a hot-head boss who fires anyone at the drop of a hat.

Maybe it's for the same reason Lucy Carmichael continued to work as Mr. Mooney's secretary or Jane Hathaway as Milburn Drysdale's. The quick-tempered boss was a sitcom staple.

Something that never made sense to me was how Angelino hired Jack after his impersonation of David Miller, most jobs especially restaurants would do a background check on new hires.

Something that never made sense to me was how Angelino hired Jack after his impersonation of David Miller, most jobs especially restaurants would do a background check on new hires.
You're joking? A sitcom using common sense? Where could you start?
Where did Jack get money to buy and open a restaurant?
How would Jack cash a check for $10k made out to Grandma Tripper?
How did Jack run and cook at a restaurant when he was always with guests?
Why does a restaurant owner, an emergency room nurse and a flower shop manager need to share a 2 bedroom apartment?
You could go on forever :lol:

While we're at it, how exactly did Jack continue to work for Mr. Angelino as "Jack Tripper"? Keep in mind that, as far as Mr. Angelino is considered, he really only knew him as David Miller. And yes, Mr. Angelino does at the end of that particular episode, say that he’s going to hire Jack Tripper but, it isn't like would recognize him?

Maybe he just really liked him :)