A Roth conversion is defined as a distribution from a traditional IRA and a taxable rollover to a Roth IRA. It doesn't matter if the cash is paid directly from the traditional IRA to the Roth IRA or indirectly by paying the cash to you which you then deposit yourself into the Roth IRA, the taxable result is the same. When doing it indirectly you have to make sure that the Roth IRA custodian knows that the deposit is a Roth conversion, not a regular contribution. The custodian's deposit forms will have a way to indicate that the deposit is a Roth conversion and not a regular contribution.
Money is fungible (one dollar can be substituted for any other dollar), so the dollars from the distribution that go to tax withholding are the same as any dollar you have in your pocket (except that the withholding is dollars held to your credit by the IRS instead of being in your pocket). By substituting other dollars (the dollars that you might have instead sent to the IRS as an estimated tax payment) for the ones being held by the IRS, you have the necessary cash to deposit into a Roth IRA as a Roth conversion to complete the conversion of the entire gross amount distributed.
If the process confuses you, it might be easier to make an estimated tax payment and complete Schedule AI if necessary, but Schedule AI is still not easy. I've had to do Schedule AI in the past to avoid a tax penalty due to income received late in the year and it more than doubled the amount of time that it took to prepare my tax return.
