These are excellent points thank you for bringing them to everybody's attention.

I'm a little shocked that people haven't brought this up until now. This is one of the main security issues that the BrowserGap on premise solution is built to address.

With on-premise self-hosting you can transfer the trust issue from our cloud-based system to your own private cloud or data center.

I'm not sure if all browser isolation vendors offer hybrid or on-premise solutions but I think most do. Menlo, Symantec, WEBGAP, LightPoint, Authentic8.