No, it's probably because websites running in Chrome might lack the ability to detect keystrokes in the background. If they did, that's a very very concerning security risk.
If they wanted to force you, they'd just disable the web app lmao
The web version isn't a standalone client like Signal, which registers as an additional device with e2e. WhatsApp web communicates with the WhatsApp app, so it doesn't work if the phone isn't connected to the internet (in early versions it had to be the same network, if I remember correctly).
I believe WA introduced a feature which allowed the desktop app to function standalone like Signal. Signal Desktop adds a second device with it's own keys, so contacts send automatically messages to two devices. I'm not sure if it works the same for WA, and if they even have the feature. I don't have a compatible desktop.