How common is it for a dino's body part to be found at any rig?

Fairly common; in fact, geological research is conducted much in the same way; wells can often be located at the same depth as many dino remains, as the pockets may have existed in the same era. I grew up around construction workers of all kinds, including those who worked on this type of structure; apparently fossils make the rounds just like any other geological artifact that might be resting at a drilling site.

I'm frankly shocked they didn't find an entire perfectly preserved T.rex encased in amber, so much that one could even call it a plot hole that they didn't find it.

hey supadude no problem. Probably could have explained it better, but it's my dad's expertise not mine lol

They're playing with you supadude.

While not impossible, it's incredibly rare to find identifiable dinosaur remains at a drill site unless you dig for them yourself. This is because of two reasons:

1. Despite common belief, 99.9% of oil does not come from dino remains. Any basic geology class will teach you that oil was created by innumerable plants and animals that died over billions of years. Their remains were covered by sediment over billions of years, and the decay of those remains combined with pressure (add confusing chemistry explanation here) blah blah blah then created crude oil. While dead dinos certainly contributed to the mass of decaying organism, they were only a very small portion. Plant life was the major contributor.

2. Any skeletal remains would be ground to incredibly small fragments by the drill bit. Think about this for a minute: the drill bits used in oil drilling grind through any type of rock and pulverize it. What bone, however fossilized, do you realistically think will make it past the drill bit?

Some dinos remains have been found at drill sites, but almost always on land and almost never because of the actual drilling.

I was a District Manager for National Oilwell Varco 2001-2008