The A-to-C cable often does not work because the resistors are supposed to be in there.
So if you are having complete charge failures, try a different cable.
A-C cable assembly always works, CC signal is connected within the cable to Vbus via 56kOhm resistor, but that's only relevant to the downstream port, not to the upstream USB-A power sourcing port which does not have access to the CC signal. Upstream port provides power unconditionally within some limits depending on port type (CDP/DCP/USB3.0/2.0 data port/...).
That's close, but it's not quite complete. There seems to be lots of confusion here, and that's natural: It is confusing.
For just-getting-power from a USB A port into a USB C peripheral: There are supposed to be 2 resistors in the peripheral device [always], and also 1 resistor within the cable for USB-to-legacy cables[1]. That's 3 resistors, total, to get a relatively dumb USB-C equipped peripheral device to reliably charge from both USB A and USB C hosts/chargers/whatevers:
The cable itself: It gets an internal 56k pullup resistor between Vbus and USB C pin A5 -- which is the CC line [yes singular]). This resistor signifies the capabilities of the host/charger/whatever for devices that care (some do care, some do not care).
The peripheral: This minimally needs two pulldown resistors [commonly 5.1k], between each of CC1 and CC2 [yes a plurality] and ground[2]. This tells a compliant USB C host/charger/whatever "It's OK! Send the juice juice!" regardless of connector orientation.
[1]: https://www.usb.org/sites/default/files/USB%20Type-C%20Spec%... section 3.5
[bleh]: Again, it is a confusing thing. Nobody said that dealing with such flexible, ambidextrous connections would be simple. CC performs a lot of different tasks: It can be a bidirectional serial bus for active PD negotiations, and/or a resistor network for passively dealing with power, and it's the bit that performs detection of cable orientation for applications where that matters, and it probably does other stuff too.
That single little wire is clever AF. It'd be simpler to use multiple wires instead of just one, but that would take more copper. Copper is expensive, and we each save a tiny bit of money (or a large pile of money globally) by using less copper instead of more of it.