Paul’s words in 1 Corinthians really are just mostly cultural.
Same with similar messages elsewhere in St Paul’s letters.
His points are based mostly on the cultural norms of the times and the specific places to whom he wrote. In those times, such things as having heads covered or not, or the length of the hair meant very specific things to those people: things that are now mostly meaningless in our own society.
Every society has clothing and grooming styles and gestures that have meanings within that culture. Today, we would immediately recognize a military or police uniform (to take some obvious examples). That same clothing would have been meaningless in Corinth 2000 years ago. We might say that it’s inappropriate to wear a football jersey to Sunday Mass; someone in St Paul’s time would not even know what it means.
We can try to understand those cultural customs, but at the same time we cannot go too far in trying to apply them today precisely because the meaning behind those customs (like long vs short hair) just have no meaning or completely different meaning in our society today.
As for clerics wearing hats in Church: it’s simple. We have our own liturgical customs, which have developed over centuries from a variety of sources. Some of them conform to what St Paul wrote, some don’t. It doesn’t matter in the end because Paul’s rules were cultural, not essential to the faith.
