I understand that, you can't search something in web app if web app stores data in an encrypted format.

It's a classical security vs. convenience tradeoff.

I barely use desktop clients for email, except for backup. It's either web app or mobile app, and here Protonmail doesn't work for me.

My use case would be a trusted non-megacorp in non-eyes country with standard security. Something like for example Fastmail based in Switzerland.