Into , but I don't know why it just sounds correct😂.

We didn’t manage to get to the new restaurant as …. (We were planning to go but failed) We didn’t get to go to the new restaurant as … (similar) We didn’t get a table/reservation at the new restaurant. (Talking about it being booked is redundant) But ‘into’ is better than ‘in’ because one of the meanings of ‘getting into’ is obtaining the ability to go somewhere desirable. That fits this case. I got into Stanford. I got into the game because my Dad has season tickets. Let’s see if we can get into the restaurant. Maybe there was a cancellation.

I would say "We didn't manage to get in at the new restaurant." "At" is the preposition we use to indicate places or addresses. When you say it this way, "in" is an adverb, not a preposition. If you said "We got in the new restaurant", then "in" would be a preposition, and the sentence would mean you got inside the physical building. That is not the thought you want to express. You could also say "We didn't get a table in (or "at") the new restaurant." That use of "in" as a preposition makes good sense because the tables really are inside the physical building.