I think there's another step you can take, that could eventually lead to having more contact with these women, although it doesn't directly focus on asking them specifically for a one-on-one chat.
My advice would be to treat everyone the same, so even the males that are usually dominating the conversations, and just shoot a message to everyone explaining what you did here: That you're new to the group and that you'd love to chat with them one-on-one because you'd like to get to know them a bit better, and that you feel the big Zoom meetings or game-nights are getting in the way of doing so. You could even drop such a message in a 'group chat' to gauge interest. Inviting everyone can't make anyone feel singled out.
I'm stealing that idea from new co-workers: The team I work on has been working from home for almost a year now, but in that time we've added three new team members and a new product owner. We have a daily 'chattiness' meeting, of about 15 minutes, but that meeting is always dominated by the same person to whine about work from home and her kids. Those co-workers could've singled out the people that don't talk in that meeting a lot, but instead they just invited everyone, including the whiner, for a chat to get to know each other a bit better, if they wanted to.
The second part is important too: Leave a choice (to accept or not). Our team has people that like to keep work and private life separate. Those people are less interested in chats like these and find the 15 minute chat each day generally enough. Leaving the choice up to them gives them a way to gracefully opt-out. In your case, leaving the choice to other people takes into account that some people may just be there for the games and the zoom meetings (those can decline or ignore your invite), while others may actually be looking for the more personal level contact (and they can accept the invite).
With the new team members, we initially planned a 30 minute window to just get to know each other a bit more, talk about our hobbies, where we live, and from there the conversation just followed naturally. With two out of those three new team members, the meeting was too short and at the end, we just planned a next chat to continue talking. We ended up combining these, and chats with already existing team members, into a single meeting, and now every Friday afternoon I hang out with four other co-workers for a little while after work, just chatting about all sorts of things (while not having to listen to whining about work from home and kids).
One last note: A MeetUp that has game nights to me doesn't sound like one where people may join because they're looking 'to get to know people better' based on gender. So it may not be naturally for the women to want to know you better too, not in the way your sentence about getting to know them implies. In other words: While the above tactic might get a few people to respond, it might also fall very flat after that.