Generating functions are often treated as purely formal objects. You only have to worry about convergence if you need to evaluate them at some complex number: https://en.wikipedia.org/wiki/Formal_power_series
Are you trying to use complex analysis to study fast-growing sequences?
If a generating function does not condense to some small symbolic expression, it is next to useless. Yes, the radius of convergence is not very relevant for G.F., however, without any convergence, there will be no shorthand symbolic expression. And that's really what we are after with G.F. For example 1/(1-x) for the OGF of the sequence 1,1,1,1.....etc. We don't care that the interval of convergence is between (-1,1). However if there was no convergence at all, the expression 1+x+x^2+x^3..... would not simplify.
Are you trying to use complex analysis to study fast-growing sequences?