I was investigating the same issue of ArcoMage HD (GitHub), an opensource React card game of mine.
The issue seems to be introduced by some very recent version (likely in Aug 2021 or shortly before it) of Android and Android Chrome (it'll be fine if you use the latest Chrome + an old Android).
This pretty annoying, bug-like behaviour makes "display": "fullscreen" totally unusable because when a user want to click a link/button, he will have to double tap the link/button! - the first tap cannot trigger the click of the link/button and can only make navigation bar and status bar appear, then the user must immediately tap the link/button for a second time, this time, the click or touch event is triggered.
Until Android/Chrome fixes this issue, a temporary fix that a PWA developer can do is: Use "display": "standalone", then either put a "toggle fullscreen" button (using Fullscreen API) somewhere, or check window.matchMedia("(display-mode: standalone)").matches and if it's true (meaning it's PWA) then make a transparent button covering the whole page, when the user clicks anywhere on the page, he actually clicks the button, which requests the fullscreen mode and deletes itself.
(Previously I thought we can use "display": "fullscreen" then enter fullscreen mode with the API when the app is loaded. Unfortunately, Chrome requires a user gesture to enter fullscreen)
For fullscreen toggling, you can use this snippet, or screenfull.js if you want better compatibility.
P.S. Chromium issue page:
https://bugs.chromium.org/p/chromium/issues/detail?id=1232956
Good news: it's fixed in Chrome 95.0.4637.0 which will be available on Canary (unstable version) in the next few days (and we'll probably need to wait a little longer for the stable version to fix it)
23 Oct edit: Chrome (for Android) stable version is 95.x now