@Evelyn @seabrig29 @DavidR @Lex @jody
I think I got to the bottom of the problem, and it’s not something we can repair in-house. The VFD looks fried. 
I don’t think it was caused by any kind of operator error; they do go bad, and this unit is decades(?) old. The bottom line is I think it’s best to call the manufacturer and have them replace it. 
It took a bit to figure this out because I went down a rabbit hole, thinking the PLC was bad. Thanks to James for pointing out that I had the Ethernet cable in the wrong jack on the PC, which made the PLC look bad. He said “I think it goes over here,” and I said “No no, it’s in the right place.” After he left, I reconsidered, and moved it over, and the red FLT lights went away and the blinking green happy lights returned. No reason to think the PLC is bad now.
That got things back to where they were: the machine fully boots up, but soon shows “088 Spindle 1 stopped error”. That suggests the VFD (variable-frequency drive). It’s what takes in the 60Hz AC, turns it into DC, then back into AC at the frequency you need to turn the spindle at the right speed.
You can probe the diode bridge on the AC input just from the terminals with a meter. Out of 6 diodes, four look fine, but two have become open circuit. Possibly that’s fixable, if it’s just a fried diode or two, or a module with all 6 in one.
However, after opening it up further, I saw some charring on the circuit board, somewhat far away from the input terminals. Also, while the remaining good diodes were producing the 340V DC, there were no signs of life. The fans should be spinning, and there should be some lights on the control pad, but there were none. So I think the diodes blew, and the power supply for the logic did, too.
Look at the charring on the white adhesive goop on the transistor mounted to the angled aluminum heat sink. The resistor in front of it let out some smoke.
That particular model (Delta VFD110V23A-2) isn’t made anymore, but I see one one eBay for $1,500. But there’s more to it than just swapping in another one. There are several parameters that have to be set depending on the spindle, and on how it interfaces to the controller. We have the spindle manual, and could figure out at least some of them, but a mistake here could damage the spindle, and that’s the most expensive single part of the whole machine.
So I think we should have the manufacturer come out and quote the repair.