Saving changes not working with php8.1 as soon as code is not basic html

[OliverFrWTFalreadyTakenComeOnNow]

Hello,

Environment:

Debian Buster. Virtualmin panel allowing to choose default PHP mode (currently 7.4.33) but also to make exceptions and have whatever's installed in a directory of choice to run another different of PHP (testing version 8.1.13). MariaDB.

Testing grounds: a brand new wordpress, installed on an empty directory with a freshly downloaded wordpress zip archive, using a brand new freshly created database. Only 1 plugin was added, Classic Widgets.
But also a clone of an existing blog (cloned both files and database, wp-settings edited to change the site’s home url) that is still running fine in php 7.4 today, but in which there is the bug I come to report when ran in php 8.1.

The bug:

Saving the text contents of widgets only works when this is basic html content. (Think p, center.)

Anything more complex, a script, a form, or php, it won’t be saved. I click the “Save” button, a spinning wheel icon starts rotating left of the Save button… and nothing more happens. if I come back five actual minutes later to check, the button is still spinning, if I refresh the page, the change hasn’t been saved.

A screenshot is attached:

An example of the code that won’t be saved into a widget area, this is, plainly, a search form, no more, no less:

<div class="search_main">
<form method="get" class="searchform" action="/" autocomplete="off">
<input type="text" class="field s" name="s" value="Search..." onfocus="if (this.value == 'Search...') {this.value = '';}" onblur="if (this.value == '') {this.value = 'Search...';}" autocomplete="off">
<button type="submit" class="fa fa-search submit" name="submit" value="Search"></button>
</form>
<div class="fix"></div>
</div>

I checked the website’s apache error log, without a risk of false positives as it was a test ground: nothing is recorded, at all, when I click the “Save” button and nothing happens.

Maybe important to mention:

I'm not 100% sure this is classic-widgets' fault. I tried with the “Classic widgets” plugin disabled , went to create a “Code” widget, pasted inside the code I quoted above, and it gave “There was an error. The response is not a valid JSON response.”. No idea if that’s the same issue or not, though.