At an altitude of 35000 feet, 600 mph works out to about Mach 0.904.
(There's a handy calculator for this here: https://www.grc.nasa.gov/www/k-12/rocket/machu.html.)
A Boeing 777 can fly at least as fast as Mach 0.89 at cruising altitudes. (Source: https://www.flightglobal.com/news/articles/flight-test-boeing-777-300er-fast-and-heavy-176585/.) It's possible you actually were flying at Mach 0.89 and the "600 mph" includes some rounding up.
Of course the aircraft is designed to fly slower as well. Otherwise you couldn't take off or land.
So yes, the engines are designed to deal with getting a lot of air in their intakes, up to some maximum amount, and also to operate while getting a lot less air.
When you drive a conventional automobile it also is able to operate on much less air than the maximum it can take.
The Mach number on the second part of the trip was only about Mach 0.76 according to your numbers. But with a tailwind, you can get more efficient flight (fewer pounds of fuel per mile) by flying slower and letting the tailwind carry you forward.
Going into a stiff headwind you want to fly much faster to reduce the amount of time you're fighting that headwind.
(This is oversimplified, because the altitude also affects the optimal cruise speed, but the optimal cruising speeds tend to be greater in a headwind than in a tailwind.)