At some level of income, I think the safe harbor minimum is 110%, not 100%. See if that applies to you and make sure you have that covered.
I would keep with the withholding method to pay up to the safe harbor amount, since that's not subject to matching the timing of tax payments with the taxable income event. So you may shoot yourself in the foot (not badly) by paying off the whole capital gain now with a one-time payment.
I'd probably just keep the cash since there's no benefit to paying the full taxes early. You've got about 8 months before you have to file and pay the rest. At 1% that's about $300, isn't it? That's something, considering you make that by doing absolutely nothing more.
I don't quite follow your last statement:
The common thinking seems to be "pay taxes as late as possible" but that means holding a ton of cash that earns nothing. If I pay off all these taxes, I can put additional free cash to work.
You don't get additional free cash to work with by making a lump sum tax payment. You are counting your larger paycheck (because you no longer would have the extra withholding) as cash to work with, but aren't seeing that you are taking a lump sum of extra money from somewhere else, which is presumably invested and working for you. Right? You wouldn't just be making a $40,000 payoff, but also an additional amount equal to your extra withholding you've set up.