Ok to normalize GDP change by population change?
I guess there might be some purpose to this but generally this is very different than $g-r$ and probably a bad idea. 

It isn't the same people growing the population and growing GDP.  $\Delta GDP$  is the $\Delta$ (output of the existing population) + $\Delta$ (output of the new population). When you normalize by new population, you get 

$\Delta$ (output per capita of the new population) 
plus a nuisance term of 

$\Delta$ (output of the existing population) / $\Delta$ population). 

Unless incomes drive positive population changes this term is not going to behave much at all like $g-r$.