Would it be legal to have spinners on the landing gear of a private aircraft?
Not on any certified aircraft. On an experimental-amateur built (E-AB) aircraft it
"may" be legal if you can convince the DAR who inspects/signs off the aircraft. Or just install them after it's signed off and don't tell... ...
If no, why not?
On certified aircraft any changes/modifications not covered in the aircraft's type certificate data sheet (TCDS). Can only be done/installed if they are covered by a supplemental type certificate (STC).
(actually some mods are done just by an IA signing off on a 337 form, but that's getting to complicated.)Basically any aftermarket part or mod to a certified aircraft has to be approved by the FAA, whether through updates by the original manufacturer , or an independent company/individual who has gone to the effort to prove to the FAA that said said change from original design is safe so they can sell it with an STC. Also no matter how great the idea, getting STC from the FAA is neither cheap, nor easy. Which is why aftermarket stuff for aircraft is so expensive, when you buy aftermarket stuff for aircraft you are not just paying for the part, but for the STC to make it legal. For example it can cost around $1,000 to change an older Cessna 100 series aircraft from a generator to a modern alternator, even though the alternator and bracket really cost less then $300. You're really paying for the piece of paper that makes it legal.