A right isn't gone when someone abuses your rights. Rights is a meta-ethical issue."Inherent" rights IS a misunderstood, cloudy concept. If they were inherent in the sense of permanence, they couldn't be taken away, but they often are.
The concept of "rights" was invented to perform a certain function in society. It's an invention, not a discovery. Someone discovered that the human is by nature, inherently rational; but nobody discovered that humans also have rights.
The argument is over what our moral obligation to animals is. I believe that we are at the point where it is moral to treat them a certain way, which ultimately leads to us bestowing rights on them. Those rights, once defined, would cascade down into laws and so forth... but ultimately all of that is an expression of society's view on the moral obligation towards animals.
Not everyone agrees yet, so it is in flux (western culture anyway), but society has indirectly bestowed certain rights through animal cruelty laws, etc.
In my case, I think civilization's ultimate goal is to eliminate suffering(*) and doing so at the expense of any living creature is negative. It's an immoral utilitarian outcome. It wasn't always, since our choices were done at the margin: before we didn't have the ability to survive without less moral choices, eventually turned from survival to utilitarian and is done for increasingly spurious reasons (typically resistance against change).
So, overall, I support it because I believe it to be moral. Regardless, it is already happening and almost certainly will be increasingly codified. Personhood is actually a poor way of addressing the issue - they don't have the same moral framework as humans.
(*) Civilization also gives security (risk of suffering) and so forth, so it's naturally a complicated issue. Advancement, security, happiness, freedom... complicated.