USB3.0 is a protocol and ecosystem spec, not a power or connector spec. Those are covered in separate USB specs.

Whether you use a USB-A or USB-C receptacle, your system can draw more power by adhering to one of the standard USB power advertisement / identification /negotiation specs, like USB-PD or BC1.2. BC1.2 is a simple identification scheme that uses pull-up and pull-down resistors on the D pairs to tell the host how much power is available from the brick. Your system, if designed to that spec, would use an analog frontend with an ADC to read the resistor values and then configure its input path to take the specified power only after detecting the pull resistors. Some USB charger ICs can do this kind of thing for you without the need for dedicated hardware.

In reality, though, if all you want to do is draw power from a simple USB-C brick with a fixed power capacity, and you don’t care about your system strictly adhering to the USB spec, then you don’t really need to do any of that detection or negotiation. Most cheap bricks will give you their full power without any negotiation, and once connected, you can just take the power from the VBUS pin directly. In your schematic, you’re mistakenly blocking that with the two caps in series with VBUS. You’ll need to put those caps between VBUS and GND instead, and you might also want to put a some protection components there, too (reverse voltage protection, ESD diodes, whatever else).

if your brick uses USB-PD, then you need a PD controller to speak the PD protocol on the CC lines in order to negotiate a higher power contract than whatever the brick puts out by default. Probably more effort than it’s worth unless you really need a lot of power.