With a toddler and it being your first dog, no. It would indeed be madness.
I am a qualified and experienced dog trainer/behaviourist. I WFH fulltime and my kids are teenagers with plenty of doggy experience.
We've just gotten a puppy and I question my sanity on a daily basis. I've just booked a two week holiday from work to dedicate the entire two weeks to transform the baby velociraptor currently inhabiting my puppy's body back into a puppy.
Toddlers and puppies are a questionable mix at the best of times but as this is your first dog it would be ten times more questionable.
I would always recommend an older, housebroken dog with some basic obedience training as a first dog but especially with a toddler in the mix. If you are dead set on a puppy wait until the toddler is in fulltime school and expect to take leave from work for training.
Check out puppy101 on Reddit for advice and an insight on real life with an untrained puppy.
If you do decide to go down the sensible, older dog route many small rescues consider homes with young children. Greyhounds in particular are great first dogs. Ditto lurchers, whippets, italian greyhounds and staffies. Avoid working breeds and as tempting as it is avoid Rommie rescues. Street dogs are rarely suited to family homes without longterm training and management.