I'm not too sure reinfection is as important mathematically as some think when you realize the goal will be to operate below a threshold of cases, not eliminate covid entirely.
The problem is that Covid-19 spreads extremely fast in unprepared populations. It only takes one unaware person to infect hundreds. Check out what happened in South Korea where they had the problem contained until one woman decided to go to church.
The woman went to a very large church, with over 1,000 people in the room. Such large gatherings won't be allowed for a while.
A few religious leaders seem intent on making those large gatherings happen, regardless of prevailing medical and legal advice.
Churches are usually exempt from the shelter in place rules. Thankfully most are still observing the rules, but it's not because the government is forcing them to.
Reinfection is only unimportant if you don't have a large number of people who are going act to reinfect. If you do have a large number of infections, then of course refection is going to matter. And we do have a large number of cases in the most affected counties.
Basically, just like you don't want to overwhelm hospitals, you don't want to overwhelm your contact tracing infrastructure
It just takes one super spreader to reinfect a population, especially without herd immunity or vaccinations.
Which is exactly why we need to aim for herd immunity as fast as possible, but not too fast to overwhelm the healthcare system.