I've been assuming this is what I'd do, but haven't looked into all the minutiae yet. So practically speaking, can I open a Traditional IRA, dump in whatever amount, and immediately convert it to a Roth IRA (or even better, roll it over to my existing Roth IRA), thus stepping around any and all tax considerations? Will all brokerages do this for me? If I want to do this annually, do I have to create a new account each time, or can I keep using the empty Traditional IRA?
Nondeductible traditional IRA contributions are still subject to contribution maximums, it's just that there's no upper income limit.
When converting Nondeductible Traditional IRAs to ROTH, any gains will be taxed. There are aggregation rules, so if you have one account with gains, you can't just say I'll open up a new account and convert on day 1, you will still face taxation on the gain in the other IRA (I can't recall at the moment if it's proportional or another method).
If you have your own business, you can set up a ROTH 401k. If you have no employees other than yourself and your spouse, you can set up a Solo 401k for much less than a regular 401k would cost.
If you qualify, you might also be able to set up a well overfunded life insurance policy. That won't allow you to invest in individual stocks, but you could invest in separate accounts that are similar to mutual funds, or if you prefer using it for fixed or guaranteed assets, you could do that. If done properly this can be quite advantageous, and isn't subject to some of the maximums that IRAs are subject to.