Hello Nikita,
Thank you for posting question on Microsoft Windows Forum!
Based on the issue description as well as the provided screenshots. Well! The possible reason the Remote Desktop Web Client struggles in private mode is that the browser restricts access to certain Input Method Editor (IME) APIs and local storage to prevent "fingerprinting" (where websites identify you based on your unique settings). Since dead keys rely on the browser's ability to buffer a keystroke and wait for the next one, private mode often breaks that "wait and combine" logic to keep the session isolated.
The suggestion here for the potential workaround is to use the Alternative Keyboard Layout Setting. Instead of relying on the IME toggle, look for the Keyboard layout dropdown in the connection settings (before you hit "Connect"). Set it explicitly to Czech (QWERTZ) rather than "None" or "Auto." This tells the Web Client to use a specific mapping table that can sometimes bypass the browser's IME restrictions by manually stitching the dead keys together.
Another suggestion is to change the Keyboard settings in ChromeOS. If you are forced to use the Web Client, try changing the Chromebook's internal keyboard setting to Czech (QWERTZ) - Unicode. Sometimes, using a Unicode-based layout on the host forces the browser to send the finished character as a single input event rather than two separate keystrokes, which the VM will then accept regardless of dead key logic.
The plausible explanation to your query of Why you can't have "Default VM settings + Host Dead Keys" in Private Mode. In fact, Under the current architecture, if the browser is in private mode, it cannot "borrow" the host's dead key logic because the browser does not trust the host enough to share that buffer. You are essentially asking the browser to perform a "stateful" action (remembering the + key) in a "stateless" environment.
Hope the provided information is helpful! If it is. It is appreciated to consider clicking "Accept Answer". Should you have more questions, feel free to leave a message.