Corion,
Just in case this can help tracking the issue that is causing chrome/chromium version 94. and 95. to freeze, I added the launch_arg option and the result is shown below:

my $mech = WWW::Mechanize::Chrome->new(
    headless => 0,
    launch_arg => [ " --enable-logging", "--v=1" ]
);
[download]

chrome_bebug.log file

[10576:11820:1029/142219.260:ERROR:chrome_browser_main_extra_parts_met
+rics.cc(230)] crbug.com/1216328: Checking Bluetooth availability star
+ted. Please report if there is no report that this ends.
[10576:11820:1029/142219.260:ERROR:chrome_browser_main_extra_parts_met
+rics.cc(233)] crbug.com/1216328: Checking Bluetooth availability ende
+d.
[download]