I've never had issues with integration tests running with real databases -- they never felt slow or incurred any significant amount of time for me.
I also don't think unit tests bring as much value as integration tests. In fact, a lot of times unit tests are IMO useless or just make your code harder to change. The more towards testing implementation the worse it gets IMO, unless I really really care that something is done in a very peculiar way, which is not very often.
My opinion will be of course biased by my past experiences, but this has worked well for me so far with both monoliths and microservices, from e-shops and real estate marketplaces to IoTs.