I play Civ 6 exclusively nowadays, but I have less than a 1000 hours in it, while I have 3000+ in Civ 5. I do empathize with the OP somewhat.
One of the reasons is I think Civ 6 throws so many small decisions at you, that the decisions that are affected by your unique abilities are few and far between. In Civ 5, quite a few decisions were automatic (you are going to build all buildings in each city. If you want a wonder, you build it in the city with most production, and unless you are doing a high-risk Liberty strategy, you are going to build 4 or 5 cities). That meant that the leader ability seemed to pop out more, and it was easier to identify what it did for your game looking back on it as well. Civ 6's horrible UI also means the numeric benefits of your abilities are also buried in the data somewhere.
That brings me to my perceived second reason. If I look back at my most recent game with Ethiopia, the decisions that won me the game were that Petra city I settled, letting a few barb camps live to get some extra city states, and using the mountain range nearby to build lots of campuses and holy sites. I only built 2 rock hewn churches; one for the era score, and one next to a volcano later (beause I was swimming in faith already, so I'd rather had mines), and also built only one copy of my UU for the era score (no clear need for war by then). The only really big change is that I really focused settling on hills. This meant that my cities were sometimes one-tile off from what normally would be ideal city placement. That didn't feel too impactful - but it likely was. IIRC one of the designers of the third edition of D&D once stated that it is better to give players one big ability that has obvious effects rather than many small ones, simply for psychological purposes. The impact should be significant and clear. Maybe that is why I too, just like the OP, sometimes feel that the civ abilities in civ 6 matter less, despite the fact that they have an effect on my game. Do I know how much faith and science I gained from Ethiopia's abilities in my last game? No, not a clue. Probably a lot, the game did seem easy, but it was a small and hidden benefit er city.No tooltips clearly tell me what was added. In contrast, I really noticed the +2 faith on my monuments in Civ 5! They got me an early pantheon and religion, and due to Civ 5's faith yield per city being quite low, that +2 faith was tangible. I also saw it clearly each time I opened the city screen.