Since you mention a browser assume you are talking about a Kindle Fire of some sort?
Anyway been reading and it depends on which Kindle Fire. Some can apparently read local html files, some cant. You apparently can send html files to your Amazon account and they will convert them to ebook format which you can then download to your Kindle. Some third party services that can do conversion online apparently.
Oh and Calibre apparently can convert html to your choice ebook format. And maybe a plugin for LibreWriter than can do this. Hope that helps. Maybe somebody with an actual Kindle will answer you though I think you will have to specify WHICH model Kindle you have interest in.
I just use the Kindle4PC app in WINE, for my needs that works, looked at Kindle gadgets but after spectacular failure of an old Sony ebook reader, my interest in expensive single purpose gadgets waned greatly. Just not enough value for money least for my needs. I think Paperwhite could be great thing especially say backpacking or where you dont have electric since it can go weeks without recharging. Far as I know the Fire has browser, the Paperwhite doesnt.
If you want to try converting html to azw3 (not sure if Kindle still does straight mobi) with Calibre locally, might try then reading it with the Kindle4PC app. Take some time but no money.
Well can save you some time far as Kindle4PC app goes. I have version 1.33 on my computer. Just checked and it allows local importation of pdf. Thats it, anything else has to come through your Amazon account. No idea why, you would think Amazon would want the Kindles and Kindle apps to be as universal as possible, but guess not. They seem to want a walled garden. Pretty sure you can transfer azw3 books to a Kindle gadget locally but wont swear to that, like said, dont have one.
They did make life kinda complicated, yes you can transfer mobi files via usb to kindle fire and other kindles or at least you could in 2021: https://www.alphr.com/how-to-add-mobi-files-to-kindle/ But like I say apparently to read local html files in Kindle Fire browser, well it depends on the particular generation and model, some do, some dont.