I have been remote for 11 years now and have two children 9 and 12. I can't imagine what it would have been like if I had spent the week out commuting between the hours of 7am to 7pm, we would have lost out on so much bonding time with their daddy being an integral part of their life. I managed to be highly productive, build a career, yet be there for bath times, tucking into bed, school plays / sports days, nursing…
I’ve also been working from home and very glad to have been there for my kids when they are young, but highly productive are not two words I’d use to describe the experience. I guess my time management and prioritization skill have improved by necessity, but the lack of uninterrupted quiet time is real.
The tradeoff is being away from the people I love for 8-10 hours a day, and even just being present in a baby's/toddler's life is huge. I'd still stay home again.
I recommend finding a nearby coworking space. Spend 2-4 hours of the house if the chaos is too much.