I understand traditional IRA contributions to be pre-tax.
They're often pre-tax, yes. After-tax is also an option. It's the only traditional IRA option if your income is too high to deduct contributions. After-tax IRA is usually not a great choice for long-term investments because the growth in an IRA will eventually be taxed as regular income, compared to lower capital gains rates if you leave your after-tax money out of an IRA.
Why/how would I add $6k ($6.5K for 2023) after-tax to a traditional IRA?
Why: you make too much money to deduct an IRA contribution or make a Roth IRA contribution, and you want to do the "backdoor Roth" contribution. Although there's an income limit to move money directly from your checking account to your Roth IRA, there's no income limit to make after-tax traditional IRA contributions, and no income limit to convert money from traditional IRA to Roth IRA. Combine these two and you have a way to get money into a Roth IRA even at a very high income.
The one big gotcha in this is that the conversion step is prorated between pre-tax and after-tax amounts. So if you have any pre-tax money at all in an IRA, the backdoor isn't really available to you.
How: instead of claiming a deduction for the contribution on Line 20 of Schedule 1, you would fill out Part I of
Form 8606 to declare the contribution as non-deductible.
I am thinking of the Roth conversion as $6k pre-tax addition, $6k conversion to roth, then tax on the converted $6k? Is that wrong?
You only pay tax on a Roth conversion for money that was pre-tax to begin with. If your IRA is 100% after-tax money, none of the conversion is taxable. But if you have a big pre-tax balance to begin with, and you add a small amount of after-tax money, then your conversion will be mostly taxable.
Would I only pay more taxes if the amount I converted (and was therefore taxed on) was greater than the $6k? What is the benefit of having a zero balance in the tIRA?
See above. The primary motivation to do an after-tax IRA contribution is to immediately turn around and convert to Roth. If you have pre-tax money in the IRA already, a Roth conversion means paying tax on that money. Move the pre-tax money to a 401(k) where it won't contribute to this calculation and you can do a Roth conversion with only the after-tax amounts where no additional tax will be owed.