Maybe if you target someone and go to an option and a trade-like window pops up with the broken equipment and the repair material, I wouldn't imagine it being too hard. But how to find who and what can repair your equipment?
A "/sea all Gld" followed by filtering the search by rank and location like what was available in XI could easily make you find who's able to repair your items. :)
I think SP gains would have to be reduced or capped somehow. Possibly even eliminated for repairs through that type of system. As it is, SP from repairs is pretty trivial, especially for low level repairs on a mid-high rank crafter. (I got 1 SP for repairing my rank 12 bronze chainmail with my rank 31 armorer the other day). But the idea of being able to actually progress your character while you're afk just isn't a smart one. People who don't craft would be screaming bloody murder the instant they found out that it was possible to rank up a crafting profession while you were afk.
I see your point here, but I really don't think the SP gains from repairing items is significant enough where you go afk for a few hours and come back and find out you've gotten through half your level already. As you said, the SP and gil gains would have to be minimal. But not non-existant. There needs to be -some- incentive for crafters to stay in auto-repair mode.
I think a better alternative to afk player repairs as per the OP would be able to make it so that retainers could repair gear in line with what the player who owns the retainer could repair. I think we'd all like XIV to thrive and prosper and if it ever came down to the point where login queues were a reality on any server at any time, one of the first things SE would need to look at is not allowing people to stay logged in when they're not playing.
I also don't think that having such an auto-repair mode would make that
many more players remain online while going afk. You see, crafters who are able to remain online if they so wished...are already
leaving their characters online simply due to the fact that they need to unload a crap tonne of items more than their retainer hold. Due to this reason, I doubt there will be any significant increase in the number of crafters remaining online. Sure there will be a small increase, no doubt, but probably not a big one.
However, your retainer idea is a great one (if it can indeed be coded in). Having your [main] retainer able to fix items that your character would usually be able to fix would eliminate the need to have your character logged in while you're afk. A certain search function would have to be implemented so that you're able to search for retainers that can fix your gear. I personally think this would be great, but with SE's [often questionable] reasonings, I can't imagine they'd do something like this.