In basic terms, to ensure that you have 3A available from the USB-C source, all you need to do is pull down each of the CC pins with 5.1k (separately), and then if either CC pin is above about 1.5v (1.7v - 8% I think) then you should have a 3A capable supply.
If both CC pins are below 1.5v, then you have less current available.
But remember that people might use a Type-A to Type-C cable on a Type-A supply and in that case, checking CC pins isn't enough, or might be entirely false. If Type-A to Type-C use is a possibility you'd need to also interrogate the D+/D- lines for that varied collection of voltages that Apple and others used.
I think the chip linked above also just deals with CC, there are probably some that will look at both CC and D+/-
Monitoring the bus voltage and shutting down if it sags (ie, brown out detection) is probably a more universal way of handling it gracefully.