Cracking the Coding Interview came out in 2008 and had a short section talking about the Google interview process, which I believe was system design + a coding problem that would probably be close to a modern LC question.

I did a fair number of coding interviews (for SRE positions) at Google (I don't actually know how many in total, but 25-30 is probably a safe guess) and, yes, it started with a small problem that should be solvable in well under half the interview time. I only used problems that passed my "is this fair to the candidate" screening (look at problem, try to write a working solution, if that takes more than 7 minutes, the problem is not fair).

The value in the interview comes from (a) listening to the candidate narrating their thought process during the coding, (ii) discussing the solution with the candidate, sometimes in terms of complexity, sometimes in terms of correctness, depending on what is on the board, (3) refining the code written (add more functionality, change functionality).

For a language I knew, I tended to overlook small syntactic mistakes (a whiteboard does not have any syntax highlighting) and if there was any questions about library functions, I would give an answer (and note down what I said, so that would be the standard used when judging) and the bulk of the score came from the discussion about the code, not the code itself.

If that matches what you mean by "LeetCode-style interview", that's certainly been in place since at least 2011. If it does not, things may have changed, but at least the aim wit ha coding interview back when I was still at Google was less "get some code judge, on the code" and more "get some code, judge the discussion about the code". It is also entirely possible that interviewing for SWE positions was different frmo hiring for SRE positions.