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We should start a boycott. :^)
Yeah, early on, there was a casual statement of "Oh we only give the Grand Companies male bird because something something babies."
The warrior of light will start to sue ishgard government to stop monopolizing the female chocobos
"Born and bred in the city-state of Ishgard, the majority of company chocobos are geldings"
Geldings are castrated males, usually a term for horses as well. It's in the actual chocobo description that we have unbreedable male chocobos. There are reasons for this in the cut scenes, mostly saying that the Ishgardians didn't trust the Grand Companies with females or breeding males as they felt that the population would get out of control (not so much in these words, but heavily implied) and they gave us geldings so that they would be more compliant.
Somehow Godbert twisted their leg however for the Gold Saucer.
Fixed that for you. ;)
But yeah, to summarize what others have said, your company chocobo you're issued with is actually provided by Ishgard who only release gelded male birds to stables to prevent their monopoly on chocobo breeding from being threatened.
The race chocobos you can breed for the Gold Saucer are only allowed to be bred and have females and ungelded males because you're only being technically leased them (you don't technically 'own' them, hence why you can only use them to race and not summon them out in the field to ride as a mount), presumably part of a deal Godbert did with the Ishgardian stables to allow breeding (of course, he might have also just thrown a lot of gil their way too, it worked either way ;)).