Yes, you are very much correct. Unfortunately, if HHMI and the like are unwilling* to do these sorts of funding experiments, I'm not really sure who will.
* I can't claim to be up-to-date on the distribution of labs funded so for all I know someone is already doing this.
HHMI is not the only one - certainly DARPA is willing to provide large funds for specific areas of interest (though a VERY different funding model). Wellcome trust, the Allen Institute, Broad institute, and others also work with slightly different models that can complement the NIH funding lines.
* I can't claim to be up-to-date on the distribution of labs funded so for all I know someone is already doing this.