Ok, I'm using Mullvad (Firefox fork) but it does work on Safari. I'll investigate further when I get a mo, thanks for getting back.
Discussion
Oh, yeah that's the one that behaves like Tor browser without Tor? A lot of things are disabled including the storage we need to save the critical lightning data.
We mention it more here: https://github.com/MutinyWallet/mutiny-web/wiki/Browser-Compatibility
Great, thanks!