I don't understand the dandelion hate. They are puffballs for maybe two weeks tops. The rest of the time they're either invisible or beautiful flowers.
Also, I'm like the only yard with fireflies in my neighborhood.
Daikon radishes. They grow in about anything and are especially good at clay busting. Grow a bunch then let them die back. Till them in and repeat until you get enough environment for the worms to take over the tilling. You can keep piling on radishes with something like clover and peas to add some nitrogen fixers. This is more a pasture revitalization technique, but if you don't mind being the weird radish guy for two or three years (depending on local conditions), you could do it on a smaller scale for a lawn
The message of the saying is that you only think that grass is greener because you're not living with it. The grass isn't actually greener on the other side of the fence
I actually have serious doubts that plants grow better in the complex soup of fats and proteins that a body turns into. In fact I’m pretty sure I remember reading that the romanticized idea of turning your body into a tree after you die basically doesn’t work for this reason!
Biologist here, the body itself isn't what grows the "greener grass". It's just the start of a long biological process that will lead to ecological growth.
A tree isn't going to feed directly off of the body. But the decay process will provide nutrients to the tree. We're talking about insects and fungus at various levels of the process here. You can look up things like the Trophic Levels and Nutrient Cycles for more details on this whole process.
TLDR would be that the corpse floods the area with nutrients and maybe even kills off the plants with over abundance of nitrogen, but then fungus and bugs move in, then bigger bugs and small animals, and so on and then better plant systems. It's kind of neat.