Does everybody replace their tires every five years

Probably not. I usually have to replace them at the mileage limit anyway, but I've definitely seen people running 10 year old tires without issue. The performance and safety of the tire degrades over time though... tires are not where I would start cheaping out on my motorcycle.

Have I been in huge levels of danger for years now, and am just lucky as hell that these things haven't gone kablooey on me?

It's not like they slough off the rims after five years, but you definitely have been in increasing danger, especially if those tires have seen a lot of sun.

Is the article I linked alarmist because they're trying to sell me tires?

Probably

Am I in danger if I put off replacing these things?

Yes, but I haven't seen any studies quantifying how much danger.

Is there any reason I should consider anything other than replacing with exactly the same tires over again?

Depending on your use case, there are probably better tires out there, specially if you're moving from bias-ply to radials. Regardless of what you pick, I'm pretty sure you're going to notice a huge difference as soon as you put them on.