every mempool is technically private, the important part is it being possible to globally broadcast and receive relatively easily
Discussion
I think the incentives for receiving easily is stronger than the global broadcast, especially if a large partition of the network begins filtering valid fee paying tx's with juicy fees.
Although the two are somewhat related.
simple thought experiment:
would you want to be reliant on a single pool to mine your transactions?
This only applies to the sending party. The receiving one is the one we have to incentivize to include the tx in a block.
To answer the question, no, I do not like single points of failure.
Need more mempools rx'ing over short wave radio.