I have been using Send to kindle[1] for years to upload epubs directly to Kindle. It even syncs the position in the books so that if I read it on my Kindle, it syncs the position to the Kindle app on my phone.
I've found that Send to Kindle does a much worse job at converting to MOBI than Calibre, which is a bit ironic but on par with Amazon's overall treatment of non-Kindle books.
Back when MOBI was still the primary format for Kindles, it would actually allow you to upload .mobi files directly via Send to Kindle.
These days it wants ePub only, but things also seem to "just work", and I always assumed it's because they added support for it on device side. What are the issues that you ran into?
[1] https://www.amazon.com/sendtokindle