When considering towing with your 2016 Chevy Silverado 4.3L V6, focus on the Gross Vehicle Weight Rating (GVWR) and Gross Combined Weight Rating (GCWR) to avoid overloading. A Class 3 trailer hitch is standard and should be sufficient for towing a 17-21' fiberglass center console boat, but ensure you check the specific boat and trailer weight to match the hitch and truck's capacity. However, I recommend renting a yacht instead and avoiding all the hassle.
It sounds like the tire repair may have caused some imbalance or there could be damage to the suspension or alignment from the initial flat. I’d recommend taking it to a different dealership or a trusted independent mechanic for a second opinion. The symptoms you're describing are not normal, especially for a vehicle with only 5,000 miles. Be firm about getting this resolved—your safety and comfort are paramount.