I think the answer to this should be:
My prefix has won the fights
The ELECT is the person/group that has won an election or vote. Re Hint 1, it literally means 'chosen'.
(Alternatively, the intent here may be a clue for ELECTRO, the name of a Puerto Rican wrestler and a Spider-Man supervillain, both of whom have 'won the fights' on several occasions...)
My infix marks the start of flights
In darts, the OCHE is a line on the floor behind which the darts must be thrown at the board. The darts 'start their flights' from here.
My suffix you can almost solve
The suffix is MISTRY, a soundalike of sorts for 'MYSTERY' - something tricky to solve (or 'almost beyond understanding', as per Hint 3).
My whole makes friendly rubber revolve
ELECTROCHEMISTRY underpins the electric power found in an electric battery. Such a battery might be found in an electric car - an environmentally-friendly way to make rubber tyres revolve...
As for the title...
Electric cars might be 'part of the solution' for tackling global warming, hence electrochemistry itself might have a substantial role to play here.