My recommendation is to have a lot of different stuff in an interview, so you aren't making a final judgement on someone over any individual part of the interview. That means the candidate can stuff up one or more parts of the interview, and you can still get good signal.

For example, do all the things you suggest. Get them to write some simple code. Talk to them about an algorithm problem. Also, give them some simpleish code with some failing unit tests and ask them to debug the code. (This requires some prep, but its a fabulous assessment to do.) Ask them about their prior work. Just do everything you can think of, and don't give any specific part of the interview too much time or attention.

In my experience, this gives candidates a lot more opportunities to impress me. I don't really care if one person on our team is particularly weak on data structures. Its kinda better if we have someone who's an absolute gun at debugging, and someone else who's amazing at data structure work. That creates a better team than if everyone is impressive in the same way.