I don't know of a site that lists all trails in the US. This is probably because trails are managed by the same patchwork of authority that manages the land. There are various federal, state, regional, town, and private groups that manage land that contains trails open to the public. The best I can advise is to pick a small area you are interested in and drill down to find trails there.
Just because there is no easy place to find a listing of all trails in the US, that doesn't mean they don't exist. You would get completely the wrong impression simply by comparing the 5k versus 250k numbers of listed trails you found. There are some long thru-trails many people will have heard of, like the Appalachian Trail, the Pacific Crest Trail, and the Continental divide trail. Then there are regional ones like the Long Trail in Vermont, the Arizona Trail, etc. But the vast majority are in local networks of trails scattered an many places.
In terms of total miles, these local trails exist in vast numbers compared to the few well known long ones. The town I live in (Groton MA) is about 34 square miles and has over 100 miles of hiking trails. You're unlikely to find them on any national web site. And, we're just one little town. Nearby towns don't have quite as many, but they all have some local trails and conservation areas. We also have a few smaller regional trails just in this area of New England, like the M+M, Midstate, Wapack, Bay Circuit, and a few others. Are any of those on your list? Probably not most of them.