I have two kids (5/7) and have always only had one car. However, we live in a bikeable/walkable area with some public transit.
I bike, bus or carpool to work and leave the car for my husband (SAHD). He uses it as needed. Whenever we need to get two different places by car, we've always been able to figure out a solution, which incidentally hasn't yet included rideshare.
I would start by paying attention to the trips you're taking now. Looks like you have the everyday dropoff/pickup taken care of, what about weekends? But don't try to plan for everything that might possibly happen; there will always be a solution.
Infant seats that click into a base are by far the easiest to install until you get to boosters, so if you had to take an Uber or whatever, I wouldn't worry about it at this age. And, really, kids can fit in those seats longer than most people use them. Mine had a 30lb weight limit, which would've lasted my kid until he was 2 (granted, he was skinny). We only moved him into a regular rear-facing seat because 25lb of kid plus car seat is a lot to lug around all the time, but that's really less of an issue if it's an occasional Uber trip. If it gets to that point, there are also carts that essentially turn the larger kind of car seat into a stroller.
Generally, it's not recommended to bike with a child under 1 year old, either in a trailer or in a bike seat. I personally thought the seats that attached to a bike felt safer, especially the kind where the kid sits in front of the parent, but there's not good data on it. There is pretty good data that, even with car seats, driving with kids isn't as safe as we think it is. The most common cause of death in children under the age of 15 is unintentional injury, and the most common cause of unintentional injury is car accidents.