Whether you get told the reason for the crash, and how useful that error message actually is for a standard user, will depend on the application. Some may just crash silently and others may pop an error that looks completely unrelated. Some may even manage to survive it and keep going, perhaps with a slight knock on performance or with some weird behaviour.
I mean, yeah, if you're putting your pagefile on HDD it's not a great idea. It's generally a bunch of small random reads which is ideal for an SSD but the complete opposite of what you want to use an HDD for.
Whether or not the program itself is running from an SSD or HDD is somewhat irrelevant though, Windows isn't closing minimized apps and reloading them to their prior state from SSD if you disable pagefile. It's keeping them in RAM and taking up space that would otherwise be used for filesystem caching.
Because you have (or had) "Shutdown: Clear virtual memory pagefile" enabled,
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In fact if you, or any "optimization" tools you've downloaded, have fiddled with stuff like this I'd probably just do a clean install rather than try and work out exactly what has gone wrong and where.