There are quite a few steps before actual stocks can change hands. For instance the seller could have sold without actually owning the stocks, it might be he even can't deliver them, in which case the clearing will have to take care of that, etc. All these steps could have been automated long ago if the will was there, the reason they haven't is not because of the currency used but because of traditional common practice and legacy reasons.
One real reason why digital currencies are not widely deployed is crime. Already there is a growing problem of ransomware demanding Bitcoin payments; a "true" digital currency that was as convenient as Bitcoin and allowed offline (peer-to-peer) payments (which Bitcoin does not and cannot support) would enable "perfect crimes." This was a concern raised in the 90s when digital cash was being proposed:
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.46...
This was one of the concerns that prevented banks from adopting e-cash (as the term is understood by cryptographers), though I have been told by people who worked on this that law enforcement agencies were satisfied that they could still "follow the money" since the anonymity property was narrowly defined (banks could not link deposits to withdrawals, but banks would still know who had made a particular deposit, merchants would still be able to identify customers, etc.).
A digital currency would make possible to settle a trade stock in minutes or even seconds.
A digital currency would remove a lot of transactions costs associated with buying/selling stocks.