A personal/private account is a tax-sheltered savings plan, like an IRA. As in Social Security, payments into an IRA must come out of wage or salary income, and are limited (though the limit is a dollar limit, not a percent of earnings). Like the proposed private/personal account, an individual (retirement) account can be invested any way the owner wants, with some reasonable limitations, and the earnings accumulate tax-free until taken out. The biggest difference is that IRA earnings are fully taxable when taken out, while Social Security benefits are only partly taxable.
IRAs and personal/private accounts are savings plans. Social Security is an insurance plan, not a savings plan. Participants in an insurance plan pay regular amounts into a common pool, from which they collect when a specific need occurs. Social Security is almost the opposite of life insurance. Life insurance pays you more than you paid in if you die too soon; Social Security pays you more than you paid in if you live too long.
Insurance plans, including Social Security, are pay-as-you-go plans. Payments out of an insurance plan come out of current payments into the plan by other participants. Only a relatively small fund has to be in reserve to cover fluctuations.
There is a good reason why Social Security was set up as an insurance plan instead of a savings plan. The Social Security Act was signed into law in 1935. Payments into the system began in January 1937. For the next five years, retirees received only small lump-sum payments, so as to build up a reserve fund (the Trust Fund). Regular monthly benefits began in January, 1942.
It took only five years from the time payments into the system began until the time full monthly payments were made to retirees. The first monthly check went to Ida May Fuller, who paid in a total of $24.75 in three years before retirement, lived to age 100, and collected a total of $22,888.92 in Social Security benefits during her lifetime.
If Social Security had been set up as a savings plan, Ida May Fuller would have received only the principal and interest on $24.75 for the rest of her life. The first retiree to receive full benefits would have had to save for a whole working career, about forty years. The first full monthly payment would have been made, not in 1942, but in (or around) 1977.
We are looking at two completely different kinds of plans: insurance plans like Social Security, and savings plans like IRAs and private/personal accounts. Savings plans, by their very nature, take much longer to start up than insurance plans. If an insurance plan is underfunded, you can't shore it up by adding a savings plan to it, and reducing the payments into it by diverting them to a savings plan (as proposed) can only destroy it.