If all the cockroaches died then other animals and some other insects would do the job of the cockroaches in dealing with decaying matter etc and even then the decaying matter would go away eventually.
Roaches are a primary food source for a number of birds, wasps and rodents and other creatures.
If you remove the roaches, the population of those creatures declines.
That might not sound like a big deal - less rats even sounds like a good thing - until you start to think about what creatures rats are a primary food source for, and what creatures those creatures are a food source for, and so on and so forth all the way up the food chain.
There's no really good way to model or predict what would happen if you removed all species of roaches from the planet.
It might be relatively minor ecological shifts, or it might be a catastrophic trophic collapse.
Better to let nature just take care her own extinctions as much as possible.
Most cockroaches feed on decaying organic matter, which traps a lot of nitrogen.
In other words, extinction of cockroaches would have a big impact on forest health and therefore indirectly on all the species that live there." In short, we really, really need cockroach poop.
However if cockroaches did go extinct then the earth could get by just fine in other ways.
Cockroaches are very disgusting insects and I really hate them and had to deal with the cockroaches in a house I used to live in.
The cockroaches were so disgusting and I could never get rid of them that I decided to just burn the old house down to rid the house of the cockroaches.
The house was an old farm house and I had hired exterminators and did everything I could to get rid of the disgusting cockroaches but nothing seemed to help until I burned the house down.
Cockroaches are pretty beneficial to the earth though because they break down organic waste and in the process, they add nutrients to the soil through their waste.
Cockroaches are also food for some mammals and reptiles.
But I can't stand the cockroaches and will get rid of them whenever I see them.