How does a ribosome gather tRNAs at a fast enough rate for Translation?
The answer is simply due to the random walk of the amino acids. Over small volumes, this process is incredibly quick and is also responsible for nucleotide delivery to polymerases as well proteins searching for their substrates, in fact, only a few enzymes/proteins are limited by the rates of random walk motion. I'm sure there are lots of papers which discuss the rates of trna movement in the cell, but this is the first one that I found: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2727733/
(You are completely right about competing tRNAs and the orientation of the tRNA entering the ribosome, all of these factors are taken into account in calculations. Ribosome animations are also a massive oversimplification, they illustrate certain points well but they are by no means accurate, they are useful for reaching purposes!)