The issue is that the 3rd party apps would have no way of knowing when you have sent a message from another device. 3rd party apps don’t have a way to do this because no other service does this. We can’t add this entire new paradigm to other people’s apps.
Before others point our Mighty Text and Pushbullet, understand they are just remote controlling your phone, no phone = no messages. Broken phone, dead battery, left at home, stolen, dunked in water and Anywhere will still work. We can continually add the features you like but the 3rd party apps can’t add this capability since they are not the messaging service provider.