The problem isn't where she'll work - in an office or at home - it's that you're both making up scenarios in your heads instead of talking about concrete fears. You're worried the kids will be neglected and there won't be food on the table, she's afraid she won't be heard. But these are just assumptions, and they keep you from seeing a real solution.
Suggest to your wife that you try remote work for a month, but with one condition - you discuss in advance how life will actually be organized. Not "she'll just abandon everything" and not "she'll be a wonder woman juggling three jobs," but specifically: who takes care of the kids and when, who cooks, how you step in during crunch moments (like when she has a deadline). I knew a guy who was against his wife working remotely - for exactly this reason. Then they mapped out a daily routine, and it turned out she actually spent more time with the kids than before because there's no commute. The key is not just to agree or disagree, but to test your assumptions against reality.
And one more important thing: if the family needs the money, this isn't just her want, it's a shared necessity. We're not talking about her whim here. Maybe the real issue isn't remote work itself, but that you both think working from home is easier and can be done on the side? Actually, it usually turns out the opposite - a home workspace often demands more discipline.