I thought the Google Assistant integration could also expose Google Home devices to HA so HA could control them but I believe I am wrong in this?
I have a device that works with Google Home but does not have HA integration/support. Is there a way to use Google Home as a kind of middle man?
Correct, you can’t do that - Google don’t provide any way of doing it.
Perhaps with IFTTT?
What is the device?
Daikin Perfera airco/heatpump
There seems to be IFTTT support.
Daikin Online Controller works better with IFTTT
So you probably can activate the IFTTT applets from HA using the IFTTT integration.
Although no longer maintained it still works perfectly, try looking in to “assistant relay” in the HACS
I had a look but it seems outdated. To connect you need a “out of home” ID, which since their screenshots of the app are very outdated, is impossible to find in the app.