Is it feasible to go uninsured and save the money instead?

My partner and I recently had a baby, and adding her to my employer-provided insurance plan (my partner is under a state plan) brings my cost to $444/month. I know that’s cheaper than many private plans, but it’s still outrageous. I’ve been wondering about the feasibility of declining my coverage and putting, say, $300/month into our high-interest savings account, which would yield something like $4,000 a year. I don’t have any expensive medications or health complications. I go to the doctor twice a year. If that holds for a few years, I could easily build up at least $10,000 in a personal healthcare fund, which I think would be enough to cover most major bills.

Right??

This all seems much saner to me than paying SO much monthly for a HDHP, which would just stick me with a high bill anyway. But nobody I know does this. Everybody says “oh you should just pay for the insurance – just in case.” In case of what? The most horrific medical emergency? Getting stuck with a $30,000 bill instead of a $70,000 bill? Either one bankrupts us. We pay for ambulances either way.

But maybe I’m just being naive? I don’t know a lot about the ins and outs of this stuff. I just don’t like being pressured into this horrible system when a more reasonable option seems attainable.