Some things would have made a lot more sense (i.e. Boromir's concern for the hobbits, Elven Cloak giving, Speech at Osgiliath, The Death of Saruman, The Witch King breaking Gandalf's staff, The Mouth of Sauron and other scenes as well) and they are far truer to the books.
Quite simply...they would have been too long.
Recall, theatres need to show these multiple times per day...the longer the movie is...the less times they can show it.
These movies clock in at:
Pt1: 3hr 28m.
Pt2: 3hr 44m
Pt3: 4hr 12m.
Factor in getting the theatres ready, preview times etc you're not looking at much change from 5 to 6 hours overall.
Assuming you can get people to sit still for that long...most won't
The average mainstream US/European movie clocks in at around 2 hours for a reason, research has shown that that is the optimal length theatergoers prefer (or can stand).
So they can rotate each movie in around 3 to 3/5 hours and the theatre can get one or two more showings in every day and that's MONEY.
Anything else is leaving money on the table.
Much of the marketing for movies and the eventual income from them has to do with the ability to capture the most dollars from the audience as possible. With the LotR franchise, there was already a proven market for video and books. Naturally, if positioned properly there was room for toys, accessories, costumes, collectibles and everything else. There will always be rabid fans who have to have everything, including alternate versions of any material if there is enough of a difference. This can be seen especially in today's comic books, where some titles will be released with alternate covers (I believe Harley Quinn #1 from October 2016 holds the record for some 120 or so "variant" covers for a single title).
Now, keep the above info in your back pocket for a sec. Let's go back to what was filmed for the 3 movies in the trilogy. The movies were filmed over a span of 438 days. They had hours of film in the can, but had to pare it down to what an average movie-goer could handle. Basically, 2 hours is considered a ceiling for most movies, you almost never want to go over that because you begin to lose the audience. So each movie was stripped down to a negotiable length and released in theaters.
Now, what do you do with the rest of the video? Well, you cater to the general public first. So the "Theatrical Release" was put out on DVD. Now, let's get back to our back pocket from above. You know you've got rabid fans who have to have everything, so a few months later you release a "Director's Cut". Then you repackage the whole thing and release it again. Then you release a Blu-Ray version with a couple extra "extras" so the hard-core collectors buy that up too. Then, maybe a video game or 2.
The franchise, in its entirety, is said to have brought in over $4B US (nearly $3B alone from box office sales).
They did. I have seen the extended versions of "The Fellowship of the Ring" and "The Two Towers" in theater in Paris. This was a national release in France (I don't know for other countries) that was just before release of the next episode and also about at the same time of availability of the dvd version.
According to the bonus of these DVDs, the extended version movie editing was ready only months after the initial release.