"Women always have people to turn to, friends, parents. A single Dad is much harder to get support for. "
Ummm, no. That's not how life works.
My DH died and there are no living grandparents to help out, my siblings are spread across the globe and whilst I have lots of lovely friends, they all have families and lives of their own, so cannot drop everything if & when I need assistance.
I've had to change my work, buy in help when needed, severely limit my social life and take my kids with me to events that I would much rather have gone to childfree.
You, by contrast, have living parents who offer free childcare at weekends. You have time to run, go to the gym as well as keep the job you had in marriage (which is presumably full time) so you are getting waaaaay more support than I ever had or will get.
I understand that you're frustrated and want adult company, but solo parenting usually means that your child(ren) have been through a significant amount of trauma leading to a wholly absent parent. So you really need to focus on your DC for a while.
My DC aren't even aware of the dates I have been on, nor should they be. If things get serious with someone I may well introduce them a year or so in.
Continue to date your GF in your free time, your DC doesn't need to know much about it, but be honest with your GF that further kids MAY be on your agenda, but not until your existing DC is emotionally secure and that could take years; potentially longer than this woman's biological clock has to offer.
Dating is fine and acceptable.
Rushing into a blended family in your situation is not advisable, and it seems the only real reason for the rush is your GF's desire to have a baby.
You might have to let this relationship go and date someone who isn't wanting DC in the near future, just so you can slow down the introduction process of a new GF to your child without ruining the GF's chances of ever having DC.