peter491 Support for DSLR cameras is complicated since most manufacturers do not open their software for third party developers. If one camera of a vendor, say Sony, is supported then it doesn’t automatically mean that ALL cameras of that vendor are supported since the software on a camera usually is very different from the software of other cameras, even of the same vendor and even from previous versions of cameras of the same line!
All DSLR camera support is done by GPhoto and ZWO adopts that in ASIAir. So, if GPhoto supports the Sony a7iii then ASIAir may too. In other words, try and find out if your camera is supported by GPhoto and let ZWO know. That way you actually help improving this great product.
ZWO state “most of the mainstream DSLRs” and apparently your camera doesn’t fulfill that criterium. There is no false advertising there. Just a misunderstanding on your side.
In the mean time you may want to point your wrath at Sony for not opening up their software instead.