First, I agree with others: you cannot afford to stay home if it means drawing from savings. You need every penny you have as an emergency fund -- particularly if you decide to remove the safety net of having a second job.
So if you want to stay home, figure out what you'd need to cut from the budget to do so. Run all your taxes -- federal, state, SS, etc. Subtract from clothes, gas, eating out, etc. to account for some of the lower costs you'd have if you were at home. Figure out how much extra the insurance would be through your DW, or if you have some sort of cheaper Obamacare option and/or state support (some states do have health insurance programs for kids from lower-income families). You need to start with hard numbers instead of guesses and estimates.
Then comes the hard stuff: you have a lot of miscellaneous stuff, like hobby spending, news access, eating out, etc. Which of those are you willing to cut to be home with the kids?
The really good news is that if you're home with the kids, you can take charge of making those cuts. You can control how much you guys spend on groceries, and how often you go out to eat, by taking over the menu planning and grocery shopping and cooking. You can control the Target spending by being the guy who goes to Target. And even better, you're the guy who likes to track spending, which is a lot easier if you're the one doing that spending. Being at home puts you in the driver's seat to save as much money as you want to, based on the lifestyle that you decide to create. It is a job in and of itself; you just get "paid" by spending less and freeing up money for other things.
One other factor: if your wife is the one with the career growth potential, you guys may be more successful long-term if you quit and take over the house-and-kid stuff, because that frees up her time and energy to focus on that career and get those promotions. There are certainly enough couples where mom stays home and dad works, and who have discovered that dad can go farther in his career than he could have if both of them were juggling work and home and kids. There's absolutely no reason why that wouldn't work regardless of the genders involved, as long as the two of you are aligned in your goals and roles, and you both dedicate yourselves to succeeding at your chosen jobs.