You absolutely should design and implement it, exactly because it is now your interface. In fact, it will add more constraints to your design, because now you have different consumers and potentially writers all competing for the same resource with potentially different access patterns. Plus the maintenance overhead that migrations of such shared tables come with. And eventually you might have data in this table that are only needed for some of the services, so you now need to implement views and access controls at the DB level.
Ideally, if you have a chance to implement it, an API is cleaner and more flexible. The problem in most cases is simply business pushing for faster features which often leads to quick hacks including just giving direct access to some DB table from another service, because the alternative would take more time, and we don't have time, we want features, now.
But I agree with your thoughts in the last paragraph. It happens very often that people don't want to undertake the effort of a whole new design or redesign to match the evolving requirements and just patch it by adding a new table to an existing DB, then another,...