Here is how to do it: https://electrum.readthedocs.io/en/latest/coldstorage.html

On an offline device I recovered my wallet via seed and then initiated a transaction. There is seemingly no way to get a transaction hex for the unsent transaction.

So the only way to get a transaction hex that I could then broadcast is to use an online watch only wallet.

Correct?

This would have made a nice bounty post. You paid 5,000 sats to boost it -- I am confident a 5,000 sat bounty would have resulted in many upvotes and many responses (and more likely, better and more comprehensive responses).

Yes. I use tailscale too beside other methods. As you said you can construct the tx in electrum, sign it, then broadcast it from other tools. Samourai, sparrow, bluewallet, sentinel nunchuk wallet apps could do that. All they have a special feature where you paste the tx code and broadcast it.

Yes, that's called "offline signing", and Electrum supports that -- even on Tails. Of course, the transaction (before being signed) needs to be created with knowledge of the UTXOs, so that part occurs on a device that is online, but your private key isn't needed for that part.

See also the following posts, found here on SN:

What Is A Partially Signed Bitcoin Transaction (PSBT)? #24779 https://thebitcoinmanual.com/articles/what-is-psbt

and

What Are Partially Signed Bitcoin Transactions (PSBTs)? #136109 https://river.com/learn/what-are-partially-signed-bitcoin-transactions-psbts

Yes, you can send from a wallet/broadcast a tx from the wallet without syncing it on your end.