Open in private window, set to offline in the console's network tab, do your thing, close the window. At least I'm not aware of a way the app could send or persist the data when you do that.
I often have the opposite problem: I'm on Firefox and try to google "FooError: Bar happened" and instead of directing to google, Firefox prompts me to select an application to open "fooerror links".
That's probably the episode they're talking about but it doesn't render their complaint less valid. It just shows the limits of streaming technology in its current form.
Finland's government-owned railway company, VR, has a notoriously bad train ticket reservation system that still uses Flash. They've only recently started to roll out more modern versions of the app (that are actually surprisingly ok quality from what I've seen).
At some point (like years ago) when browsers started dropping support for Flash, we got some nice hacky solutions [1] from tired consumers because VR themselves weren't stepping up their game. At some point I even saw VR promoting that project on their Twitter as a solution when someone complained that they couldn't book tickets. They should've paid the author and just put it on their site.
I had to use Adobe Connect recently and they also had the option to install Adobe Connect as a desktop app that seemed awful lot like it was just Chrome/Electron with Flash preinstalled... Also no Linux support, which was fun to notice just before the meeting.
I didn't read parent comment as "Microsoft and Valve are bad and I don't want to give them a dime, ever". I read it as "I want to direct 100% of my donation to the project I'm looking to support, Krita". Which sounds reasonable.
People have suggested extensions here, which is fine, but Firefox also supports this natively via custom css files that it can load for you on startup. They can be used for both styling Firefox itself and for modifying websites' styling.
https://superuser.com/a/319322/1173126
(be sure to read the comments of that answer too, these days you need to switch a flag in Ff settings to enable this feature)