A thin oil like 0w-20 may not provide adequate wear resistance in an engine, where it is not recommended. Similarly, in case of old engines, engines based on old tech., etc. where they are running fine on 10w-30 or 10w-40, switching to 0w-20 can bring harm and leakages, so, not recommended.
When engines age, gaps between different moving parts might increase, however small it is (like in microns, etc., but its there). So, if an engine is working fine on a particular engine oil grade, then no need to mess it up. Let it working on it. If problem arises, then switching to further, slightly thick oil can be a temporary solution, but thin oils like 0w-20 is not the solution in any case; at least to me.
IMO, 5w-30, 10w-30 are very safe engine oil grades, to be deployed in most of the recent cars in Pakistan, for all-around-the-year use.