... OK @Dave-H, I think I have solved this difficult riddle of yours, so I'd kindly ask for "preferential treatment" for the rest of the year ; joking aside, I bet you 100 quid you have toggled the following pref
chrome://flags/#enable-experimental-web-platform-features
to "Enabled", to squeeze more out of the old Chromium 86 JS+CSS engine, have you not?
On an epiphany (pun intended, it's already Jan 6th here), I remembered a discussion inside the Supermium issue tracker, where Client Hints was the topic... The aforementioned pref, in versions of Chrome ranging in the mid-to-late 80s, enables an alpha/gestational form of the CH API, which was only finalised in the early-to-mid 90s (and sometime after Cr87 it was put behind its own flag, decoupled from experimental-web-platform-features) ...
If you visit
https://browserleaks.com/client-hints
do you get under "Client Hints JavaScript API: API Support" -> True ?
If yes, that's the origin of your VM issue; I can replicate your 360EEv13.5 VM issue myself if I do enable that pref ("Disabled" in a fresh/default profile), please see below:
The behaviour you're experiencing is caused by Violentmonkey (and Tampermonkey ?) code but in defence of the VM maintainers, they couldn't possibly author code to cover ALL cases of users messing with the "you're-not-supposed-to-touch" Chrome's "Experiments", aka "chrome://flags/"; more so, when they're currently supporting Cr 61-131 (and each of those has its own, unique, set of "Experiments") ...
Technical stuff:
The culprit commit is the one below:
https://github.com/violentmonkey/violentmonkey/commit/d2dda25d91e3718e5b20ba4648e82baf730e6178
from May 2024; this one assumes the browser comes equipped with the final iteration of the Client Hints API, which isn't true for Cr86/87; no CH API (the default in Cr86/87) doesn't break VM >= 2.19.1, but a draft/RC CH API does ...
Mitigations:
If you want to install and use the latest version of VM in 360EEv13.x (Cr86), you'd have to
a) disable/never enable "chrome://flags/#enable-experimental-web-platform-features"; this will reduce, somewhat, web compatibility with certain sites that expect recent Chromium-based web engines...
b) if you opt to enable "chrome://flags/#enable-experimental-web-platform-features" for webcompat purposes, then you should disable Cr86's draft CH API via the cmdline switch: " --disable-features=UserAgentClientHint" (e.g., add that to loader's "Parameters"); some recent websites which rely on sniffing UA CHs may break...
If the website of interest remains functional ONLY when detecting some Client Hints (e.g., some sites "take you for a bot" and outright block you when no CHs are sent), then you'd have to enable "chrome://flags/#enable-experimental-web-platform-features" in 360EEv13.x but opt for an older version of VM, one < 2.19.1 (crx4chrome should afford those...).
Well, that's it ; detective work concluded successfully: case filed!
Best wishes ...