paradust
For storage, a bare minimum Minetest world needs 1GB, grow-able to multiple gigabytes. I don't think IndexedDB is meant for that.
The network issues are specific to Firefox. Multiple threads in Minetest do network I/O. In Chrome this works fine, but on Firefox only the main thread can do I/O, and so network requests are proxied to the main thread (this is done transparently by Firefox, using a mutex internal to firefox). Unfortunately this arrangement leads to a deadlock given the particular way it is used.
The UX issues do seem to be stubbornly persistent though. I ran into the same issues long ago porting games with NaCl and asm.js. I'm not sure what is preventing progress there.
There's an updated launcher. It's not finished yet, but it is nicer for exploring single-player community-made games: https://minetest.dustlabs.io/ez/
I'm not actively working on this, as I have concluded it is not (yet) possible to create a good web experience. There are problems with storage, network, keyboard side effects (e.g. Ctrl, W), concurrency, and performance that need to be addressed by browser vendors. I highly recommend installing the native Minetest!