16AWG cabling is a definite requirement for anything you're looking at pulling more than about 150W per cable. If "the Bitmain PSU" means the 1600W thing they're selling, it's only got 18AWG wiring so not rated for the 300W loads you might see with some miners. Probably okay for running S3 off a single cable per board but not if you want to overclock. A good 750W server PSU should handle two overclocked S3, a 1200W would handle 3 easily and just going up from there.
One benefit from using server PSUs with breakout boards is user-defined cabling. You're not limited to the wires the manufacturer thought you'd need. You can get what you want, and there are plenty of options even within this forum for wire gauges, lengths, termination styles and such. And then in the event that your PSU fails (which server PSUs tend to be more sturdy and reliable than consumer-grade ATX, though there are a few very solid ATX models with comparable reliability and efficiency), you need only a replacement PSU because the breakout board and cabling are all still good. If you have a spare PSU on hand already, it'd take about ten seconds to swap it in instead of however long it'd take to undo the octopus of wiring from an ATX and reconnect the octopus for the replacement. Makes things a lot easier.