"Is UPS offering next day service in a regional zone with a letter rate of $5?" - Some Amazon's yes, but by far no.
"list of prices for a 2 lb pkg going by UPS, FedEx Express, and FedEx Ground and Ground was much cheaper" - UPS Ground and Fedex ground will be identical everytime and Express will be a lot more.
"I guess if they are charging the same rates as UPS Ground then UPS has nothing to worry about" - I do see the logic in that. But it doesn't explain the whole story or tendencies. But your point is if UPS Ground and Fedex Ground were very near equal, why would a shipper pick a 2nd rate operating company vs UPS? Well they do, even when it is close to even. Now not all and not even most are choosing Fedex Ground, as one can see UPS Ground is something like 10x the size of Fedex Ground. But there are customers that were burned in their view by UPS on something, think claims. There are many others that look at the delivery success metrics and see Fedex Ground is comparable even with contractors vs Union. Some of it has to do with Union, that works both ways, Fedex Ground won't get some business because they aren't Union. UPS is more flexible with pickup times simply because they have more boots on the ground to do so. Many companies today do this "UPS GROUND" - "FEDEX AIR", using the perceived leaders in both segments. I get that to.
But the best overall bottom line I can give you is Fedex Ground Service is with all it has against it... still comparable in getting a package from A to B vs UPS.
But UPS or Fedex at any time could be cheaper than the other, there is no "it's a given Fedex Ground is cheaper than UPS" if it were Fedex Ground would be growing even faster, perhaps that is why they are expanding as fast as they can, so they can do that someday. Because one is right with their models, Fedex Ground should be able to under cut UPS, capacity must be the issue and if Fedex Ground is near full now, why cut your price, right? Now one can say USPS is cheaper than either Fedex or UPS, that is true.
I still feel the regional carrier is the segment that can upset the two majors and they have the contractor model and lower overhead in management, by far. The more warehouses large companies place, the more regional carriers play a part and that is happening.