You compare the blockchain to the Internet, but I think this is wrong. The blockchain is an implementation detail (a very important one, but just implementation), while the Internet is just the idea. The implementation of Internet relies on TCP/IP. So, as I see it:
Bitcoin = Internet
Blockchain = TCP/IP
And, no, we shouldn't talk about Blockchain (which will create confusion with Blockchain.info, which is kind of brand now).
I agree with you. Saying 'put it on the blockchain' is even more confusing. 'Use bitcoin' is fine. I also think blockchain.info should have called itself something else as it is confusing, though I guess it was a smart move from a business perspective.
What about coinbase? That's a key term in bitcoins... it's smart to align your business with key terms, etc...
But I had the same thought as the OP - why not move toward the "Blockchain?" I think the term "Bitcoin" is partisan at this point - as it will solely be referred to as a currency. It's confusing to new developers - those who can help out with the Bitcoin economy don't realize there's a powerful, underlying protocol.