You are applying the product rule incorrectly. You have to specify which variables your derivative holds constant "at the beginning". In other words, you don't mix what is held constant. So if you want to hold volume (and number of particles) constant you do
$$\left[\frac{\partial(PV)}{\partial T}\right]_{n,V}=V\left(\frac{\partial P}{\partial T}\right)_{n,V}+P\left(\frac{\partial V}{\partial T}\right)_{n,V}=V\left(\frac{nR}{V}\right)+0=nR$$
To verify:
$$\left[\frac{\partial(PV)}{\partial T}\right]_{n,V}=\left[\frac{\partial(nRT)}{\partial T}\right]_{n,V}=nR$$
or you can do the same thing with pressure being held constant. You will find the same thing.
What you propose will actually end up adding $nR$ twice and will thus give you $2nR$. The expression $\frac{\partial}{\partial T}(PV)$ isn't very good to use since you aren't specifying what you are holding constant (and you can't hold all other variables constant or else you wouldn't be able to change the temperature due to the ideal gas law).
Now, we technically don't need to hold $P$ or $V$ constant, but in general we would need to at least specify how one of these changes with respect to the other variables, since there is more than one way $P$ and $V$ could both vary as we change the temperature. However, it turns out for the ideal gas that this specification is irrelevant. Let's see why by only holding $n$ constant. Then we have
\begin{align}\left[\frac{\partial(PV)}{\partial T}\right]_n&=V\left(\frac{\partial P}{\partial T}\right)_{n}+P\left(\frac{\partial V}{\partial T}\right)_{n} \\ & =V\left(\frac{\partial (nRT/V)}{\partial T}\right)_{n}+P\left(\frac{\partial V}{\partial T}\right)_{n} \\ & =nR-\frac{nRT}{V}\left(\frac{\partial V}{\partial T}\right)_{n}+P\left(\frac{\partial V}{\partial T}\right)_{n} \\ & =nR-P\left(\frac{\partial V}{\partial T}\right)_{n}+P\left(\frac{\partial V}{\partial T}\right)_{n}\\ & =nR\end{align}
So as you can see, this will always be the case. The key is that you need to specify the constraints on your variables before taking the derivative. The constraints don't just come out of nowhere.