Why does the addressee on my messages turn red?

You should still check Settings > Messages > Text Message Forwarding on your iPhone to make sure your iPad is still enabled there.

I assume you have checked the obvious things like verifying that your intended recipients haven't turned off their phones, signed out of iMessage, or switched numbers.

Probably because your recipient is not signed into Apple's iMessage system and you do not have Message Forwarding enabled on your iPhone.