I believe type C can work without muxing, but only in one plug orientation.
Shorting the two lanes at the plug turns the unused lane into a length of unterminated cable hanging at the end of the connection, which is unlikely to do any good for signal integrity. May not work at all, may not work at certain cable lengths, etc - I have never tried it. Or it may be connected to a second transceiver in a type C host. This is not a problem for the USB 2.0 pair, because it is shorted at both ends, but SuperSpeed pairs are not.
Shorting would only work with single-lane A-C cables, maybe.
Type B is what you are supposed to use on a device if you don't want type C.
You can technically use type A and availability of cables is not a problem, they are widely used for some non-compliant 2.5" disk enclosures and for non-standard board-to-board PCIe (not to be confused with PCIe over thunderbolt). But it's not compliant and not idiot proof - if such cable is plugged into two hosts, one will end up powering the other's 5V rail.
Standard compliant A-A cables without Vbus connection are intended for using the USB debug capability and relatively rare. Practically all A-A cables you will find at "consumer" suppliers have Vbus.