The essence of the public option is that a person less than age 65 can enroll in Medicare. How much below age 65 and how much would the premium be? The answers to these questions are typically deemed unimportant details. However, the adage "the devil is in the details" applies. The Biden page above includes multiple answers – “premium-free” for some and tax credits for others, which in effect reduce premiums. Full-blown, the answer is a hodge-podge due to state variations and subsidies via Medicaid or other government assistance (existing or per future legislation).
Something to keep in mind is the premium that people now covered by Medicare (or Medicare Advantage) pay. The basic amount for 2020 is $144.60 monthly. Higher income people pay as much as $491.60 monthly (IRMAA, link). That is for Part B. Most people pay $0 for Part A, but a few pay something ($252 to $458 monthly).
Imagine that Joe Biden’s plan were implemented. Mary Doe age 62 takes the public option and qualifies for paying a $0 premium. Three years later when Mary Doe qualifies for Medicare, her premium will jump to the Part B basic amount, now $144.60. That could be quite a shock for some people, so it’s reasonable to expect that Biden’s plan would contain some “transition rules” (probably complicated) to ease Mary’s burden.
More people going on Medicare, with the CMS’s low reimbursement rates paid to medical providers, will put upward pressure on prices for any private or employer-paid insurance – mostly that provided by employers. For inpatient services, private payers and employers recently paid prices that were 231 percent of Medicare reimbursement, while for outpatient these entities paid 267 percent of Medicare reimbursement (link).
Since employees generally pay around 20% of the cost of employer-provided coverage, the employee cost will go up as well. Then they might choose to switch to the public option. And very likely as in Bernie Sanders’ and Elizabeth Warren’s proposals, the government would confiscate the employer’s “cost savings” when the employee opts out of the employer’s health insurance plan. With rising costs, more employers could drop their plans, pushing all of their employees toward the public option. All that is part of the unspoken future reality of Biden’s plan – push more and more people into government-run healthcare. In other words, fast-crawl socialism. Since “socialism” is vague, with different meanings to different people, better and clearer is “fast-crawl authoritarianism.” Biden’s authoritarianism is more institutional than personal, but nevertheless authoritarian. To him government knows best and is the preferred means for anything regarded as "social."
Sleepy Joe himself says nothing at all about most of the above to the general public. As far as I know, he has said nothing about taxing employers the way Sanders and Warren did. The Biden plan linked above says nil about it. Maybe he doesn’t even understand "his" own plan. Many politicians' foresight is very short-range. I’d guess that Sleepy Joe himself is not even the author of "his" plan. Some staffers wrote it. I bet he knows less about what’s in “his” plan than Nancy Pelosi knew about the Affordable Care Act (Obamacare) when she said, "We have to pass the bill so that you can find out what is in it."
Sanders' and Warren's Medicare for All would in effect force everyone to take the public "option" (in quotes because there would no option.) Of course, there is no assurance whatever that Sleepy Joe, if elected, would not quickly substitute some form of Medicare for All for the "public option" plan he promotes while campaigning, especially if pushed by Congressional Democrats. He is an accomplished flip-flopper. He might even follow the pattern of Bill and Hillary Clinton. He could appoint Kamala Harris -- who advocated Medicare for All while a presidential candidate -- healthcare czar.