In theory, yes. In practice, it's not happening. Collaboration is a very difficult thing, as you can appreciate, with each organization having its own independent board and purpose for existing.
There are little pockets of co-operation. There is an insurance company providing small charity insurance policies. The precedents are there, but we need co-operation at scale, and scaling it takes resources. That's what this fund could invest in, for example, without coming to the Treasury Board and asking for money.