It makes sense to me and is roughly what I did for roughly the same reason I think you're doing this...to avoid pro-rata rules.
I don't think you will trigger pro-rata rules and the IRA->401k rollovers should be nontaxable events.
If your 401k is not at Vanguard, Vanguard will probably require that they send you a check when you do the rollover. You will just endorse the check and send it on to your 401k custodian. It always seems nerve-wracking to me when this happens, because you've got large checks floating around in the mail, but it has always worked out just fine when I have done things like this.
Things to watch for:
1. Make sure your Schwab IRA is the only IRA you're left with. I'm assuming this is the case - that is, that you only have 3 IRAs total and not a fourth one somewhere else you've forgotten to mention.
2. Make sure your 401k allows incoming rollovers before starting this whole process. Not all do. You can ask your 401k custodian or check the summary plan document (SPD).
3. I don't think it is strictly necessary, but when I did something similarly I did the IRA->401k rollovers and made sure they completed before I did the tIRA->Roth conversion. You want to avoid some sort of administrative delay or paperwork holdup that screws up your plans.
Finally, you might as well open your Roth IRA with Schwab if you're happy with them. That should make your tIRA->Roth conversion simple.
Good luck.