"should the software show that they are eligible for a $6500 max Roth IRA contribution?"

For 2023, the total contributions you can make to all of your traditional IRAs and Roth IRAs can't be more than: $6,500 ($7,500 if you're age 50 or older) or your taxable compensation for the year, if less.

A contribution is New money. Conversions = moving around existing money.

"so that they are eligible for Roth IRA"

That's meaningless, now that they took all those actions.

For the tax year that they first put that money into Roth, that was the contribution for that tax year. Then, they recharacterized it, changed the account it is in, but still as part of the total limit for that year's for contribution. This is more like a correction to an error.

Conversions don't count against contribution limits. They  can put in $7k, and convert $500,000, for example.

So, they already made their contribution. The only difference between if they had left it in the Roth and conversion, is that their conversion might be a pro rata taxable event. Otherwise, the conversion is not deductible how they did it the first time, nor how they recharacterized it.

Their end goal is accomplished. They don't get to make another contribution for that same tax year, unless it was not as much as they qualify to put away?