This looks like a really cool project. Have you used it, and does it work nicely? Can you use it to send Lightning payments to anyone, or only to other accounts within their system? The biggest obvious downside here is that it is entirely custodial. Would be great if you could instead run your own wallet on your phone, and instead simply send and receive transaction data and the like with their service via SMS.

You are confusing airgapped wallets with broadcasting transactions. It is of course entirely possible to run an airgapped wallet and transfer signed transactions to an online computer in order to be broadcast. What OP is asking is what to do if you do not have an online computer in order to broadcast your transaction. In this case you need to use something like radio (as I've discussed earlier in this thread) to send your transaction to someone who does have an internet connection and can broadcast your transaction for you.