Whenever I play someone other than Germany I’m constantly hitting the district cap
Sounds like maybe you're not growing your cities fast enough. With domestic trade you should be growing plenty fast. Are you hitting you pop limits, or are you below the pop limits? I tend to find myself in GS with more housing than I expect and rarely hit pop limits until getting into the 15+ pop range, at which point it doesn't really matter. In terms of districts, in most cities you really only
need like 3 districts.
Granary +2, University +1, Lighthouse +1, Dam +3, Aqueduct +2, Gurdwara +1, Farm, Pasture, Camp, Planation, Fishing boats, all +.5, in addition to policies, etc.
So, assuming a river city you can easily get to pop 8 in the Medieval era with just a Granary and University, and really with improvements you'll typically be around pop 10 or 11 by then. I often also take Audience Chamber, as I think its overall the best of the first government buildings since it provides a constant bonus that is always active throughout the game and becomes even more useful as the game goes on and you get more governors. That means that in your main cities you can easily have pop 14-16 in the Medieval era, without even doing extra stuff like building an Aqueduct.
If you make an Aqueduct and Dam, now you're at pop 18-20.
But really, the main thing is getting up a Commercial Hub/Harbor, Campus, and Industrial Zone. You may also need a Holy Site.
You only need a pop of 7 for those 3 districts. On some coastal cities its nice to have a Harbor + Commercial Hub, so maybe you'll want 4 districts faster. In games where you build lots of Holy Sites, you often will use Work Ethic, which can delay the need for a IZ, so again there are really 3 main districts. But really anything past pop 10 starts to get irrelevant, because in most cities you rarely have need for more than 3 districts and even less need for more than 4.
I can't say that I never hit a district limit, but its rarely any kind of issue. I view the additional district of Germany as an extremely minor bonus, that rarely comes into play in GS. Is it ever really game changing? I don't think so. When you compare it to Spain or Cree's trade routes, or the Macedonian bonuses,
The other way to think about it is this: You really need 3 to 4 districts in most cities, maybe 5 and rarely 6 in a few. What matters is being able to build those districts ASAP. What really drives your ability to build out districts is population. The real key is not how many districts you can build per pop, but how many districts you can build by turn x since the founding of a city.
Civs like India, Indonesia, Khmer, Cree, Spain, Kongo, etc. all have bonuses that help them grow population faster and in some cases larger, which means that they will be able to build more districts faster. In the end, more population is better and more productive, so I find that fast population civs, like those mentioned have far better economies and much higher production than Germany, which has nothing to help it grow faster. So yeah, Germany can build more districts with a lower population, but in terms of being able to build X number of districts by X turn from settlement, I find that fast growing civs are actually able to build more districts faster than Germany can.