I am a little confused by multiple things in your last post, maybe you can clarify?
A thunderbolt cable was all that was needed
You will need more than just a TB cable.
The 2019 MBA has TB3 and no TB 1 or 2 ports. You would need the bidirectional adapter as well as the TB cable for this method.
Found this out after buying a usbc / a cable and thunderbolt 3 to 2 adapter (hopefully seller will take the adapter back).
Are you saying you tried the the bidirectional adapter with a USB-C to USB-A cable?
That would be combining two different methods and definitely wouldn't work.
Apple says the USB-C to USB-A cable should work, but I highly recommend using their suggestion, and not some random cable you find. Did you use this cable?:
Apple Store Search Results
www.apple.com
Like I said earlier, the TB 2/3 bidirectional adapter and a TB would definitely work, but is more than twice the amount of the USB-C to USB-A.
Some other things are weird about your post:
even though my system was one of the last 32 bit mid 2011 imacs.
The Mid 2011 iMacs are 64 bit and the iMac line had been 64 bit for years before the Mid 2011. I am not sure why that even matters though.
Do you mean you were using 32 bit SW on your iMac that is not compatible with your newer Mac?