Instead, I would like the ability to combine the routing of payments, with the traffic delivering the protocol itself.
[~snip~]
Has any work been done toward making such things possible?
The issue with this idea is that it stems from an oversimplification in basic explanations of the Bitcoin and Lightning networks. You could add a signed transaction or a channel state commitment in a low-layer protocol, but it won't be possible to 'combine' it the way you think.
I ignore Bitcoin on-chain payments in this context, because you'd need to wait for a few minutes to at least get one confirmation; by then, the HTTP request (or TCP connection, whatever protocol you like to integrate with) will already have timed out.
Lightning payments aren't 'routed' like IP packets; they're actually sequences of channel state updates in a series of channels that connects a sender to a receiver. Therefore, you can't integrate the 'routing of payments' into the 'traffic delivering the protocol itself', unless all network devices that pass along the 'low-level protocol traffic' have Lightning channels between each other.
Hence, what you propose is not possible, and I'm not aware of any work trying to make it possible, by forcing network devices to own Bitcoin and open a channel to each other network device they're connected to. Does this make sense?
Yes, makes sense. Would be an interesting experiment, but not compatible with current internet architecture. Though I suppose Internet routing could be augmented with lightning channel information, meaning links with a parallel lightning channel could take advantage of that fact, and those without such a channel, would not---but that's way above my pay grade.
I actually think SMTP postage over Lightning would be worth trying. Imagine getting paid to read (or at least receive) marketing emails! Sorting emails by postage would be an alternative to all the machine learning spam filtering. It would also reward email users for the cost to their attention, and provide a natural funding mechanism for mail servers.
The main difficulty would seem to be establishing enough of a network effect that it could get traction. If receiving mailservers could tap into lightning and get paid to receive emails they're already receiving---that could be an enticement to participate.