Well the states didn't build them, they benefitted from the same thing; wealthy men who bequeathed their estates to a state, who also happened to be fans of the gothic revival.

You'll find that the bulk of these castles were built between 1880 and 1919. This is primarily due to the confluence of three factors. First, the gothic revival of design was in full swing around the world. Many socialites were clamoring to connect with roman and english ancestry via the grand and sweeping nature of stone castles and usage of arches in contrsutction. Second, folly construction. That is, folly construction is the act of building homes and other structures with outlandish, unecessary, and sometimes gawdy features that transcend their function. Towers and ramparts are great examples, which in the eras before gunpowder were built for utility. This might be the first example in america of "yeah I know I don't need it, but it looks cool" aside from that yanky doodle fellow who insisted he was dressed in the style of macaroni. Third, wealth. The late 19th and 20th century saw a massive growth of wealth. You combine these elements and you have a collection of wealthy individuals who want to build "american castles" without thought for their function or utility. Why on mountains? Vistas, views that stretched miles, plus in the american northeast the mountain ranges are in the 2,000ft-ish elevation, which translates to about 10 degrees cooler in the summer in most places.

So wealthy guys who own large plots of land, with beautiful views on top, where it's naturally cooler in the summer months, in an era where building you own castle was a style option.

Why only in the NorthEast? I would note that most of the castles were built with stone quarried from the immediate, or nearby areas. edit - I'm having trouble finding the book I was looking for, but summarizing from memory the northeast was (and still is mostly) abundant with fieldstone and lime. Fieldstone construction had two benefits, first being that it was abundant and easily mined, and second most farmers wanted it removed to expand cultivation areas for farming. Lime was used to improve the quality of mortar (workability or making it "buttery smooth"). Again, these were locally available.

Some other examples to your list

Gathland State Park, Maryland has a castle-like monument dedicated to journalists who died covering war (the only one of its kind).

Gillette Castle, Connecticut was built by a wealthy actor (no relation to razors). Two interesting notes about Gillette. First, all of the doors are hand carved and unique. Second, the castle has multiple secret passage ways that were specifically designed by Gillette so that he could slip away from parties or make a grand entrance.

*Adding more links