Does openstreetmaps have a proper cross-platform phone app with turn by turn instructions while you are driving that works well - and does it have a good majority of costa rican roads?

Yeah, I think it is, in fact, because Waze is very easy to use for everyone, especially for people with non-technical backgrounds. The support for Spanish locale must be another reason.

Note however, that I haven't used OSM myself, so I can't truly compare them fairly.

Voice turn by turn nav and traffic updates and reporting cops, roadwork etc. I use it even on my commute just to get those alerts.