I had some wonderful experiences about rentals in Denmark. I tried both car-sharing from Copenhagen Airport, and a traditional rental out of a very small airport on a peninsula.

The car share was awesome. I got a brand-new Mercedes CLA, 200 HP petrol engine luxury coupe. I checked if I can hop over to Sweden with it, and apparently I could. The pickup was an app, I just had to call so that my account and licence was approved for driving in Denmark. Took me longer to find my way out of the garage than the whole process. It ended up costing me around 60 EUR, but if I would have brought my own car the trip would have been nearly the same, because the car share company paid the Øresund bridge toll for me.

I then got stranded later at this tiny airport in the middle of nowhere. There was a chain rental place there (maybe SIXT?), with a big friendly guy attending it. Again, took me 15 minutes from finding the office to sitting in the driver's seat of an Opel Mokka. And I don't have a credit card at all, and my debit card is weird because it's a bit different from all the others in the EU on account of it being from the NL.

I am just saying this since it sounds like a PITA to rent in the US then, it's totally different from my experience.

That said, I do rent quite a few times, but mostly car shares for small trips, mostly between big cities. Costs me 10-50 EUR a time with fuel and the insanely expensive parking included, depending on if I go between cities or not. I would need to do a lot of that for even a small car to make sense, I would need to rent a ton. I'm looking at a new job that would need me to commute internationally (admittedly it's only 250 km) almost daily, and it still seems like a train pass makes more sense, it's better with time, I can sleep or work while travelling, and it's a third of the cost of even a small car.

Again, this is not bashing, just saying how different it is between over there and over here.