Why is it that when I travel to certain places I need to ensure my passport has at least n months before it expires? So what if it's due to expire next week if I'm only staying until next week. Even if I'm staying 2 weeks and it expires tomorrow, why does that matter? I guess I might not be allowed back into my home country, but that should be my concern, not the worry of the immigration of the country I am going to.

What kind of illegal immigration / criminal activity does a country prevent, or economic benefit / any other advantage does a country get by enforcing this kind of rule?

My guess is that it's because of emergencies. If you get injured and can't or shouldn't fly home then you need more time on your passport. It is also much easier to send you home if you over stay your visa if your passport is still valid. Also the system is setup to give you a visa of a specific length (eg 30 days), they can't just give you a 2 day visa.

Also if your passport lasts for 10 years you've known when it's going to expire for quite a while, they're just expecting you to be responsible.

It's a risk mitigation measure. The three or six month requirement usually comes from how long you could stay, not how long you do stay.

You are also assuming a point-to-point trip. While a citizen of a country usually cannot be denied entry to their own country, any countries you transit are under no obligation to allow you through on an expired passport.

tl;dr filters out people who may be problematic to deport.