I would bring a stroller, but bring one that can easily fold up and moved out of the way. I have 3 kids, youngest is 6 and we've been going to Taiwan almost yearly for 15 years. We no longer need a stroller but had one or two on all our trips when they were younger. They do take up space and can be a bit of a nuisance in the crowds so sometimes the smaller umbrella strollers are best rather than the larger ones, we've used both.

Elevators are available at most (but not all) Metro stops. At some of the transfer stations it can take longer to find and take the elevators and there were many times, elevators weren't available, out of service or inconvenient. However, if that was the case I'd just carry the stroller up the stairs and continue on.

Many places sidewalk aren't great and you'd be walking in the streets. Stroller would help keep the kid from running out into traffic.