Digital money definetely eliminates the paper money involved in the system. But would it make the system completely cashless is the main question. Well, it can definetely bring in a "less cash" drive than the cashless one. Offcourse achieving a cashless society is the bigger goal but we need to wait for that for a while. For now, less-cash society is whats being offered with the advent of digital money, and the whole digitization process alltogether.
Offcourse the cryotocurrency will add to this in a good and substantial way as it is based completely on digital money, created and existant over the cloud. This if regularised in a more advanced and accepting way, will definetely pave ways for cash-less economy and will eliminate problems assiciated with paper money like theft, laundering, corruption, etc.
There's definitely quite a bit of merit to what you said, and I want to echo the idea behind your question on whether it makes any sense to go completely cashless. What if we did not have cash? Wouldn't that be some sort of restriction on society itself? Completely ridiculous. There currently is a drive to go cashless because it is a trend, it is convenient, and there is a market race between payment processors to become the most established within their respective markets -- like, see PayPal, Square, Payoneer, Amazon Payments, Visa/Mastercard/AMEX, etc.