A couple of points occur to me.
First, as you point out, whether a family can “afford” to become a single-income family depends on the choices they make about consumption. To a signficant extent, it’s a choice that is driven by the couple themselves rather than something imposed upon them externally.
As you point out, for nearly all couples choosing to become a single income family involves limiting consumption. And, if enough couples make that choice, we have a significant overall drop in consumption in the economy. And, right there, we have the makings of a recesssion.
Would unemployment go up? Well, it depends on what you mean by “unemployment”. If unpaid homemaking is an employment then, no, unemployment would not go up (or, at least, not solely because of the numbers withdrawing from the labour force to become full-time homemakers). If, on the other hand, you only count paid occupations as “employment”, then the total numbers in employment in the economy would decline signficantly and, by that measure, unemployment would go up. But if you only count those who are seeking a paid occupation as “unemployed”, then those who have withdrawn from the paid workforce are not unemployed and their withdrawal does not, of itself, increase or reduce unemployment,
If consumption were to continue unchanged, then employers would seek to replace all those who had left to become homemakers. Unemployment would fall to very low levels, wages would rise and (probably) immigration would rise. But, as already noted, consumption would also decline. Would the loss of productive capacity caused by the withdrawal of workers be offset by the decline in consumption? I have no idea which effect would predominate.
There would certainly be a lot of disruption in the economy. The workers withdrawing from the workforce would not be the ones producing the goods and services no longer in demand, so many of the remaining workers would have to redeploy. This wouild take time and cost money (but of course a large-scale withdrawal from the workforce of this kind would probably be spread over years, if not decades, anyway).
There would also be political consequences. If the American people as a whole came to attach a high value (or a higher value than at present) to full-time homemaking, they would, for example, expect that to be reflected in the tax system. Why should not a non-working spouse, for example, not be able to transfer tax allowances and deductions to a working spouse, for instance, so that a home with income of, say $100,000 would pay the same tax regardless of whether that income was earned by one spouse or by two? The result would tend to be a shifting of the tax burden from married people to single people, with consequent alteration in people’s behaviour (i.e. the incentive to marry would rise).
And of course the tax base would be being eroded by the declining national payroll, so
expect to see (a) new taxes, other than income tax, and (b) a rise in income tax rates, and © pressure to curb government consumption. If the productive capacity of the US economy declines by, say, 20% due to withdrawal of workers from the workforce, can the US still afford (say) its defence commitments?
Which raises another possibility; the government might seek to encourage workers to remain in the paid workforce by (say) lowering taxes on earned income and raising them elsewhere, subsidising out-of-home childcare for working parents while reducing support for non-working parents, whatever. In doing so the government would be putting productive capacity ahead of individual welfare and maximising the autonomy of the citizen but, hey, the tax base is threatened! Desparate measures are needed!
All in all, it would be disruptive. People as a whole would certaingly be less well-off materially. The would enjoy increased welfare, however, if by “welfare” we mean the ability to make and implement lifestyle choices, unless the government intervened to try to influence their choices.