If you see a lot of old movies, the piano music played in dance halls had a strange sound-almost like a harpsichord sounds. Was this because of the cheap upright pianos of the day? You don’t seem to get this sound from modern pianos-is it because of the construction of the old uprights?
Just put thumbtacks on the felt-clad hammers that strike the strings and you get that sound.
I hear your idea…sounds good!
I should add that following Floater’s idea, while it does work, is bad for the piano and if left for too long can result in the strings breaking.
The answer is that pianos take a lot of maintenance and need to be tuned regularly. Some notes use multiple strings to sound, and when they get out of sync you do get an off-sound. The sound you hear in cheap honky-tonk pianos involves a little of the strings being out of tune and also the hammers needing new felt, causing a harder surface to strike the strings. And if the environment is warm and damp it can cause the frame of the piano to warp, which also affects the sound.
All of the above is why a lot of people are getting rid of their uprights and getting electronic keyboards instead. The modern electronic ones are quite sophisticated - the sound can be very realistic and the keys touch-sensitive, and you don’t have to worry about all the maintenance. This makes me a little sad, but I can understand the practical reasons behind it if you’re not a professional musician needing a proper instrument to practice on.
I would hypothesize old pianos, poorly recorded - if you set up a few mic’s for the scene and record the piano that way (I have NO idea if that is how they did it), you are liable to only pick up the brighter frequencies of the piano and lose some lows, which would result in that type of tone.
We definitely had a few “clangy” pianos at my high school. There were a couple of good pianos, which only the teachers and prescreened and approved piano players were allowed to play, and a few clangers which any student could use. The felt was mostly worn off (they clanged worse in the middle, where most of the play happens), and while they were regularly tuned with all the others, there just wasn’t money in the budget to restore the pianos that the teachers never played because they were clangy, so they never got unclanged.
To this day, “Heart and Soul” just doesn’t sound right to me without a little clang. 
Yep…the best example of a “clangy” piano was in the “Godfather” (1972): that piano that played while the Corleone gang war was raging-it had that classic “clang” sound.
Another word used to describe the sound is “tinny.”
Many of the turn of the last century’s pianos were fitted with a mandolin rail - a drape of felt between the hammers and the strings - which created the popular sound.
Now I’m thinking about Klondike Kate.
The piano in our church was just recently serviced to get rid of that “clangy” sound. The cause? Mold and spider webs mostly, built up on the hammer and strings.
Shame though. Many people said they like the twang. 
<nitpick> As the rest of your post sort of explained, pianos destroy themselves by just being pianos. So, I’m not sure it’s that detrimental. You just add the cost of strings to your piano repair bill 
</nitpick>
And since the question has already been answered, I’ll add that adding the rivets is usually called a tack piano, and this is my favorite example of it. I thought it was a distorted electric piano until Wikipedia came along.
Related question: many years ago, my uncle bought a player piano. It came with a ton of rolls, many of them decades old. At the time he bought the piano, rolls were still readily available-they were made by a company in Australia (called “QRS”).
Are player pianos still big in Oz?
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Shame though. Many people said they like the twang.
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Is this piano in Texas? 
As for player piano rolls, QRS has been at it since 1900 and is still in business.
Not to mention you can get several different piano sounds in one instrument. Including that old “clangy” sound if you get tired of the Grand Piano.
>sigh< There are times I still miss the battered old upright I learned on, but my e-piano is so much more practical: no tuning, much smaller footprint, easier to transport if necessary, multiple different instrument sounds…
The only downside is that it needs to be plugged into the wall and a speaker to work.
It is not unheard-of for a piano to be deliberately “tuned down” to acheive that honky-tonk sound; especially for country or folk performers. I suppose it takes fully as much skill to tweak a piano just enough to get that particular sound as it does to tune one normally. I’ve no idea whether some kind of formula exists for this kind of tuning or if it’s done by ear.