Why T-rex had tiny forelimbs

Short answer:
T-rex (Figs. 1, 2) forelimbs were former wings, not former grasping, predatory tools. Kiwi wings (Fig. 3), which have claw tips, are good analogs. Tyrannosaurus forelimbs were relatively smaller and likewise useless. Taxon exclusion is once again the reason why this has not been able to be documented before. 

What others say:
Science Daily – “The tiny arms on the otherwise mighty Tyrannosaurus rex are one of the biggest and most enduring mysteries in paleontology.”

Thoughco.com – “T. Rex males mainly used their arms and hands to grab onto females during mating (females still possessed these limbs, of course, presumably using them for the other purposes listed below).  T. Rex used its arms to lever itself off the ground if it happened to be knocked off its feet during battle,  T. Rex used its arms to clutch tightly onto squirming prey before it delivered a killer bite with its jaws. they were exactly as big as they needed to be. This fearsome dinosaur would quickly have gone extinct if it didn’t have any arms at all.”

Popularmechanics.com – “The simple truth is that scientists aren’t sure exactly why T. rex’s arms are so short, but there’s a number of possible explanations. Perhaps the most likely is that the dino’s arms just weren’t very useful.”

FieldMuseum.org – “One of the big mysteries about T. rex is its tiny forelimbs,” says Pete Makovicky, Associate Curator of Dinosaurs. “We don’t know how it used them. But there could be clues in the fossils. When a bone is used a lot, the wear and tear cause tiny fractures that heal over time. With the right tools, we can see microscopic changes in the bones caused by that healing process. You also see things like a wider bone marrow cavity. When we remove SUE’s arm, we’re going to take it to the Argonne National Laboratory to try to look for these characteristics that will tell us how much it was used.”

Chicago Tribune
story here. Great images of a rising T-rex cyber model here from Kent Stevens, U of Oregon. Another set of images here from TyrannosaurTuesday.blogspot.com.

The answer, as usual here, comes from phylogenetic analysis.
Distinct from prior tyrannosaur studies, the large reptile tree (LRT, 1040 taxa) recovers the feathered, winged theropod, Zhenyuanlong (Fig. 1). as the proximal ancestor to the tyrannosaur clade. Theropods with large feathered wings don’t use them to grasp prey or mates.

No other studies
found large feather wings on tyrannosaur ancestors. And as long as they don’t, they’ll keep thinking T-rex forelimbs were primitive grasping organs.

The kiwi forelimb is a good T-rex forelimb analog
The kiwi (Fig. 2) has vestigial forelimbs that are essentially useless. Even so, they were relatively much large than T-rex forelimbs. Kiwis have no trouble getting up, mating or anything else tyrannosaurs are supposed to do with their forelimbs.

A recent lecture
by Tyrannosaur Chronicles author Dr. David Hone, available here on YouTube, noted two traits common to all tyrannosaurs: fused nasals and D-shaped (in cross-section) premaxillary teeth. Zhenyuanlong does not have these traits. Thus the Hone traits have not been validated by the LRT. Instead those traits appear to have a wider distribution by convergence, like the arctometatarsals we looked at earlier.

Remember
we never want to put all our trust in just one or two traits (see above). Otherwise we’d be pulling a Larry Martin, famous for arguing phylogeny based on one or two traits alone. Instead we’re always looking for a suite of traits based on a character list of at least 150. The LRT has 228 multi-state characters and it continues to lump and separate all of its 1040 included taxa successfully, while documenting gradual accumulations of derived states.

In the present ancestry of tyrannosaurs other traits emerge.
Among the 228 traits, the LRT found several dozen shared by T-rex and Zhenyuanlong, including the elevated orbit, the hourglass-shaped quadrate, the pubic boot and a very short dorsal vertebral series. Noteworthy, all of these traits, other than the hourglass-shaped quadrate are also found elsewhere in the LRT by convergence. Don’t forget, we’re looking for the most parsimony in a suite of traits. Otherwise Pinnipedia and Cetacea would still be valid clades.

Essentially,
T-rex is just a giant, flightless Zhenyuanlong. No longer small enough to fly, the feathered flapping organs of T-rex became smaller due to lack of use. Blame it on the genes that those useless forelimbs keep appearing. We’ve also seen vestigial traits in pterosaurs (manual digit 5, ungual 4, pedal digit 5 in derived taxa), snake precursors (legs) and in baleen whales (tooth buds in embryos).

References
Hwang SN, Norell MA, ji Q and Gao K-Q 2004. A large compsognathid from the Early Cretaceous Yixian Formation of China. Journal of Systematic Palaeontology 2(1):13-30.
Lü J and Brusatte SL 2015. A large, short-armed, winged dromaeosaurid (Dinosauria: Theropoda) from the Early Cretaceous of China and its implications for feather evolution. Scientific Reports 5, 11775; doi: 10.1038/srep11775.
Osborn HF 1905. Tyrannosaurus and other Cretaceous carnivorous dinosaurs. Bulletin of the AMNH (New York City: American Museum of Natural History) 21 (14): 259–265.

wiki/Tyrannosaurus
wiki/Zhenyuanlong