Your math is a bit off. If you have a 14k GVWR trailer that weighs 4,500 lbs, that leaves 9,500 lbs legal payload. Quite a bit less than the 13,000 you listed.
The GVWR is the most that vehicle is rated to weigh, at any time. Even alone, unhooked. A lot of companies account for tongue weight by building the trailer to handle more than just the axle capacities added together, but not all do.
For example, my 16’ Diamond C has two 7k axles, but has a GVWR of 14,900 lbs. The sticker on the side lists my trailer weight at 4,500 lbs. I’m not sure how they got that number, because it actually weighs around 5,160 lbs, empty, as it left the dealership. Sone if that increase is likely from the HD ramps I optioned. It weighs even more, now, since I also carry chains and boomers.
In hindsight, I should have got the 8k axles. I didn’t for a few reasons, but the main one was I didn’t want a gooseneck and the highest rated ball mount and ball I could find was rated for 16k. Going to the 8k axles moves the GVWR to 18k, on these trailers.
The reason I didn’t want a gooseneck was because of the types of jobs I was doing, at that time. I wanted to keep my flatbed clear, to carry attachments, bricks or other supplies, etc. Since then, my business has adapted and changed some. I’ve only used the bed two times in ways that would have interfered with the gooseneck. It would have been better to get the gooseneck and the 10k super singles, but those weren’t an option, at that time. There are other drawbacks to the super singles, but that’s another discussion. Another factor was a goosenecks tend to cut more of the corner, in forward turns, where bumper pull tends to track closer to the towing vehicle wheels. This was important, because most of my customers are residential single family homes. Of course, nothing is free. I‘ve paid Hell trying to get my bumper pull into a few places where a gooseneck would have been much easier.
If I was buying one, now, I would get the biggest I could, axle-wise, on a low-profile. It would be a gooseneck with a deck on the neck. But, that’s not what I have, so it’s not what I use.