What do you do when PermaCast Hand Rail Anchors Have Corroded Badly- Failed in Slab?

Call around/check craigs list. I had a deck and had to install a rail. It was cheaper to call and have them drill the holes in the concrete then to rent a machine.
Yes, you'll have to drill them out. It's pretty straight forward. They have a concrete cutting drill/borer that cuts a nice clean round hole. They'll replace the anchors, concrete them in then and replace the rail.

You may want to consider replacing the railing at the same time. They make the covers/escutcheons oversized to cover the repour. Not sure on the railing but I recommend the one that the two poles that connect to the ground be connected. (That's what I will do on my replacement.) Otherwise they then to shift/twist.

anchor: Amazon.com : Replacement Fiberglass Anchor Socket for In-Ground Pool Ladders and Rails - 1.9 Inch : Swimming Pool Ladders : Patio, Lawn Garden
escutcheons: Amazon.com : PermaCast PE-0019-S Handrail Garden (not sure of those are oversized or not)
railing: Amazon.com : S.R. Smith DMS-102A-VT SealedSteel 3-Bend Braced Deck Mounted Swimming Pool Handrail with Matching Escutcheons, Taupe : Patio, Lawn Garden (Railing style that is connected)
 
If it's aluminum it should be pretty easy to drill out I'd say. How about drilling it out, and then using a rotohammer, drill a new hole a little larger and re-setting a new one? Depending on the size of your new hole, you could either use an epoxy, or a nonshrink grout like hub 100 which is really strong and easy to work with.

Or you could hire it out to a professional
 
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