Why shouldn’t people be allowed to have many children?

This is a strawman argument, but I'll bite anyway.

Two reasons, logarithmic growth, and finite resources.

Saying humanity is the virus is the part I don’t agree with because most of the destruction is caused by a very small portion of humanity.

It's not all the virus particles that kill you, just the ones that knock out your essential organs.

The biggest climate impact the typical person can have, is to have fewer children. Hands down, that means fewer cars, less farming, less production, not only for that child, but any children they may have and so on.

Having many children as a form of insurance isn't required in nearly as many places, that's old-school semi-racist thinking. Providing a better chance for children to survive, and having a society that takes care of its elders eliminates this "need" entirely.

We have enough resources on this planet to make sure all 8 billion (and more) of us can live comfortably

It's a cute idea, but getting those resources distributed evenly takes even more resources. I've read the article and it smells like bullshit to me because it uses GDP per capital to equate quality (or excess) of life. The availability of food, fresh water and shelter is not necessarily linked to GDP.

The problem in this world aren’t poor families with 5 children, it’s the wealthy elite trying to maximize their profits above everything. THEY are the virus.

Totally agree, but let's not forget, those wealthy elites are the same species. Even if most of the virus isn't killing the host, it's still the same virus as the infection that is.