Your idea is good! In the Termux-X11 plugin, this was optionally used and is selected via Preferences. The lower Nav-Bar was also used as X11 screen. On AVD sometimes a flicker bug occurs and together with the Termux Float Plugin is also often switched. Four pictures with a pixel (AVD emulator):

• shot1 shows the preferences where both choices: extra-keys and fullscreen choices are on. In the middle, Termux is in the Float-Picture without extra-keys and below the non-floating On-Screen KB
• shot2 is the X11 screen with some XClients and the KB (here with Extra Keys). The Float-Picture is then moved upwards. And the transparent Nav-Bars above and below appear only with sweep from top or bottom.
• shot3 is like shot2 but without KB (switch off by Back-Key). The Float-Picture is then moved down (last position).
• shot4 shows the screen, when tap on the Float-Picture, which is then the KB and both Nav-Bars are displayed and reducing the X11 screen. After another tap on X11, shot2 or shot3 is displayed again.

If you want to repeat this, you can use long tap to move and scale the Float-Picture or iconize (left in border). The rotation (Landscape) shows the transparent two Nav Bars at the top and right (usable areas in X11):

In the current Termux-X11 repo, a few bug fixes and features have gone lost. The current version has more disadvantages as advantages to the 3.5MB initial version