Definitely Bandcamp. I have used it a lot to support artists I like. However, the way I use Bandcamp I never think of it as a music streaming site. I usually download my songs and listen to it offline. On SoundRat I support continuous music streaming while browsing the site with the music player in the footer.
I also would like to experiment with some new ways for artists to connect with their supporters that is not possible on Bandcamp right now. Supporters can send a personal message to the artists when donating and leave comments on songs. I think there are some more "perks" that I can support in the future.
Comments on songs are not currently supported by Bandcamp, but I think their implementation of comments make sense: users can recommend songs at the album level rather than per song, so that all rave and rants about a particular album's contents can be found together rather than pieced apart in each song's download page. SoundCloud chose to go to an even more granular level and allow comments to be placed within a track's duration, and then also comments below each track. Both are interesting methods of offering a community.
Another form of community-building is what Beatport did, which encouraged remixes by offering stems and integrating with software that the artists used.
Another indie music site is good news; I remember fondly finding some good music on Amie Street before Amazon gobbled them up.
I would encourage you to check out the Bandcamp mobile apps. They allow the exact functionality you describe, of "continuous music streaming while browsing." You may be focusing on users like yourself who primarily interact with a desktop website, and download purchased music to a computer. I would urge you to consider the mobile-first use case, as this is how the trend in the market is going.
I also would like to experiment with some new ways for artists to connect with their supporters that is not possible on Bandcamp right now. Supporters can send a personal message to the artists when donating and leave comments on songs. I think there are some more "perks" that I can support in the future.