May be going a little outside the envelope here... but good formation training may not be a bad idea either. The wingman/lead relationship, situational awareness, decision making etc... seems like it produces many valuable 'airmanship' skills.
To be fair, there are LOTS of types of high performance flying that produce "valuable airmanship skills" -- formation being one of them -- but just aren't feasible for the vast majority of FAA pilot training.
If I were king for a day, I'd love to make every pilot go through the USAF's training program, which includes acro, instruments, formation, and low level navigation all within the first 5 months. That produces a VERY capable basic single engine instrument/commercial-level pilot. But, we knot that's just not really financially doable, nor is it something that provides enough value at the PPL/CPL level to make the risk and expense worth it. The cost of the first half of USAF pilot training when I went through in the late 90s (which included all of ground school and about 80 hours in the T-37 covering 6 months) was on the order of
$200,000 per person. Obviously the cost/return there just isn't possible for 99% of Americans.
I think, however, that basic acro (loops, rolls, etc) and advanced handling (REALLY unusual attitude recovery, accelerated stalls, spins, etc) are "value added" types of training at the PPL/CPL level -- ESPECIALLY for anyone operating for hire at the CPL level.
My proposal would be something on the order of aerobatics and spins being required training for the CPL (either CMEL or CSEL - a pilot would only have to accomplish the training once). Although it would require a specialized (and more expensive to operate) aircraft, as well as some different personal equipment (chutes), it would actually provide very important confidence building and skills that could save lives and equipment.