My startup is seed funded with a very small (<5) team. I am the CEO.
I feel extremely torn and am unsure what to do. In an ideal world I would take a lot of paternity leave, but I feel like I really can't take any. I just don't see how to fill the gaps that my taking time off would leave.
I'd love to know what others here have done in the past.
You have two important partnerships- the extremely extremely important one with your wife and child-to-be, and the extremely important- less so than with your family but still extremely important- with your team. Both are going to need attention and support.
All relationships- marriages and teams- are unique and there are lots of templates and patterns that work for some and not for others. My suggestion would be to start with your wife and work together on what she is going to need and what of you she can spare- and then go to the team with- here is what I will be able to bring, let's talk about what else this team is going to need. The team is small but still larger than the pair in your marriage and there may be more capacity, capability, generosity than you are currently aware of.
I personally have been in multiple of these places- on a small team with a leader who was needed elsewhere (for triplets!); in a wonderful marriage with an incredible capable wife who wanted both lots of babies and to succeed at work and needed me for a time to lead at home; which then took me away from teams that needed me but then were able nevertheless to succeed with more targeted contributions.
Prioritize the relationships, not the work. Good luck.