I am currently trying Wallabag and like it so far. Being able to self-host it is definitely a plus. I also like how clean and simple it is compared to how bloated Pocket has become, but it does not have the official integration with Kobo. It looks like there is a way to use Wallabag by installing KOReader. I will look more into that if I decide to stick with Wallabag long-term.

Brave uses their own index. Qwant and Ecosia have partnered to build their own index as well.