Agreeing and amplifying your first point: it's worse than that, it's ($12 + incremental expenses [expressed hourly]) / (1 - marginal tax rate). You have to use marginal rate, not average rate here, of course. Incremental expenses (commuting, wardrobe, lunch, and other work-related costs) are surprisingly higher than most people realize.
For a couple in even a middle marginal rate, where the second worker would drive only 20 miles round trip [including to/from daycare site twice] ($10/day) and pay $5 more for lunch than eating at home and pay an extra $1000/yr for a work wardrobe, it's easy to need to cover $4000 just for the expenses of working, plus the ~$26K or so for childcare, so you need to earn $50K just to break even if your combined marginal bracket (fed + state + local + FICA + Medicare) is "only" 40%. It's even worse if the second full-time job requires an additional or better[more reliable] car, paying for parking, or any other recurring expenses. If the stress results in needing extra spa days, meals out because everyone's too pooped to cook, vacations to get back family time, etc, it can really sky-rocket.
In terms of time, you save two adult trips to daycare, the full commute, and all the coordination overhead of dealing with childcare provider issues (family daycare providers get sick, take their own vacations, and in our case, randomly closed on short notice sometimes).
My wife is highly educated and left a very good position because she would have otherwise been working only for her 401K contributions. Several years in, everyone in our family is thrilled with that choice, it's way less stressful overall, and I'm convinced our children are way better off for it. (Though our daycare costs and marginal rate were somewhat higher than the figures in this example, there are many cases where the economic cost to one spouse staying at home is quite low.)
On your second point, I don't agree that the tax system needs to be patched for this as a special case. IMO, we should reduce taxes across the board, but failing the will to do that, we should accept that the system creates these incentives and educate people as to the math and the reality that, with more than one kid pre-K, it's almost surely a low (or even negative) cost for one parent to stay home.
Taxing a stay at home parent for imputed childcare costs is a crazily slippery slope. Must I pay taxes if I do my own dishes, clean my own house, cut my own grass, do my own laundry, cook my own dinner, fix my own brakes, clean my own gutters, fix my kid's broken toy, etc, etc?