Why is Safari asking me to allow Adobe Flash Player for unexpected sites (e.g., MacRumors, PayPal)?

For what it's worth, most of the time that you see Flash on a site like Paypal, Amazon, etc., it's related to website analytics.  If you dig into the code, this kind of content tends to be small (like 1x1 or 5x5 pixels), and these techniques in general have a lot to do with the criticism aimed at Flash around memory and power consumption.

It's not uncommon on media sites -- particularly where there's a whole host of companies involved in the metrics, ad distribution and auditing ecosystem -- where there may be dozens of these kinds of trackers from multiple parties in the ecosystem on the same page.  Since advertising pays for an insane amount of the content generated online for mass consumption, all of that auditing and tracking is necessary to ensure that content producers can make a living.

While HTML5 and JavaScript give developers similar (and in many instances, far more powerful capabilities) for these purposes, Flash Player is handy, particularly when you're talking about targeting older browser (e.g. IE8), which otherwise may lack the scripting capabilities used for creating meaningful insight into how people are using your website.  It wasn't an explicit design goal of ours to facilitate this kind of stuff, but there are a lot of smart people in the marketing space, and it's a de-facto practice at this point.

So yeah, these kinds of SWFs are probably legitimate, but also in many instances not strictly necessary to make the site work.

I've experience this as well. In the past couple of months, it seems I get this prompt ten times a day from all sorts of websites that seem to not use Flash at all anyway. Maybe they do, but like you say, I don't see any adverse effects at all from disallowing it.

Apple disables Flash Player by default in Safari 10.  If Safari is prompting you to Trust (allow) Flash Player plug-in on a specific page it would indicate the page contains some Flash content.   You're free to allow or not allow.  If the page doesn't display correctly you can go to Safari > Preferences > Security and modify the plugin settings (Enable Adobe Flash Player for Safari)

Reference:

Next Steps for Legacy Plug-ins | WebKit

Safari in macOS Sierra Deactivates Flash and Other Plug-ins By Default - Mac Rumors