Congrats! But yes it is bad, you cannot put the commute burden all on your wife. You have equal or greater responsibility for childcare moving ahead because your wife has essentially all the load, no matter how hard you think you are trying, until the kid hits 3rd party childcare. Even when you think you are doing it your wife will have a mental burden of wondering if you are doing it right.

Not to say you can't shuttle the kid by bike, we certainly did. Trailers, child seats, trailabikes. I took both my kids to their daycares/preschools by trailer and trailabike until they started K. And I could take the kids off my wife's hands by taking them for 3 hr rides/naps in the trailer. And that does entail choosing a daycare by whether you can fit it into your bike commute. But people do choose daycares based on how they fit into their car commutes so it's not too different.

There is a rule of thumb that infants shouldn't ride in a trailer until they can have the core and upper body strength to sit up by themselves and hold their head up (w/ a helmet on it) for the length of the ride. So there is a gap there between the age daycares will take infants and the age they meet those strength criteria that a bike is not an easily defensible transportation choice.