What's the rationale behind a switch to an AC transformer - would that work better with a current sensor?
Specifications of the current sensor are for AC use. Simple and cheap way to open / close a switch.
Also, what sort of wireless contact switch do you mean - something like a Zigbee door/window sensor with a magnetic contact breaker?
Any Zigbee / ZWave / WiFI modded with Tasmota device sensing 2 wires either open or closed.
IE: very cheap to take apart one of those small wireless magnetic contact switches and connect two wires out to the current sensor and it has a tiny footprint. Typically it is a little reed switch inside there that a magnet trips.
or
Purchase a Zigbee / zwave / wifi sensor with two screw terminals on it for a contact switch.
Very tiny and easy to modify and cheap to modify.
How would that connect to the current sensor?
Two wires from the current sensor (NC / NO) to the two wires on your Zigbee / ZWave or WiFi modded with Tasmota trinket.
Look at the picture above with the current sensor. It is a two wire terminal there and is open normally and closes with current thru the AC low voltage wire from the doorbell. There is a little set screw on the current sensor to adjust when the switch closes. Easy to tweak.
The mechanical chime that I have (Byron 776) has an in-built 8V transformer so wouldn't need a separate power source.
No. You want an autonomous AC transformer around 18-24 VAC (as mentioned above) that you can use for your Doorbell camera and a mechanical doorbell chime.
I had an 8VAC transformer and chime in place and while it worked fine with the Ring Doorbell it did not work with the Hikvision doorbell.
I would have no idea what to do with an ESP01 :0(
Would the current sensor + wireless contact switch be a simpler solution for someone like me?
Yes.
The world is going toward
MQTT. I am only using MQTT with Home Assistant today. It is not on the internet and only locally use it. That is where you take your el cheapo Chinese wifi switch and install Tasmota, Espurna, ESPHome on it replacing the ESP firmware on it so it doesn't talk to the cloud.
Edit: the puzzling thing is that I can trigger the Sage sensor if I connect a 12V DC transformer directly to it. I just can't work out why the full setup with the doorbell won't do the same.
Don't use the Sage sensor. You really only want to power up the doorbell and maybe the chime with the AC transformer. The current sensor is much less invasive and doesn't utilize any power to work. It gets triggered by AC current flow only. My ELK 930 current sensor did not work with the Hikvision doorbell so I left it disconnected and went to using the passive current sensor.