Just some random thoughts..
Who made SDI the arbiter what types of dives can be done solo? My agency "allows" solo cave diving for full cave divers but not intro or apprentice. I have a full cave card. I also have an OW solo card. The scope of the solo training does not include overhead or technical diving. Like someone said, the training is aimed at OW divers with about 100 dives, it is not aimed at technical or cave divers with many hundreds of dives.
Which agency are you referring to?
Does their course book explicitly state that it allows solo cave diving and does it teach solo diving techniques?
Who did you do your OW solo training through?