I want to use a power regulator PCB I designed to switch down 60V to 12V and 12V to 5V in order to power a USB plug and use it as a phone charger. My only worry is that the switching regulators I am using (TI TPS54560) can provide up to 5A of current whereas the plug is rated for 1A. Do phones cap their charging current at 1 or 2A on their own or is this something I need to implement in my system in order to avoid damaging them? Thank you for your help!
Yes, phones do cap their charging current. The cap depends on "charger signature" provided through D+/D- USB lines from the charger side, and on the state of charge (SOC) of their internal battery, plus consumption of internal CPU/display and current state of their operations (sleep/active etc.).
If you don't provide any signature (leave D+/D- floating), phones will cap their consumption at about 500 mA.
And yes, you do need to "cap" your power delivery in accord with certain I-V curves that are detailed in BC (battery charging) and PD (power delivery) USB specifications, which can be found on usb.org website, in "developers" section.