Call at First
 

Batter/Runner hits the ball to second baseman. Throw to first is dropped by first baseman near the foul line in the path of the batter/runner. Batter/runner steps into fair territory around the first baseman tags the base and falls about three feet towards second base off of first base. First baseman turns and tags the runner off the bag.

The call?

For what? You could have obstruction on F3, but I don't see anything else for the umpire to decide.

If I understand you correctly, F3 was in the way and that's why BR did this dance. In that case I've got obstruction resulting in the player not holding the base (which means I'm stretching a little to make the obstruction between 1st and 2nd). Dead ball, BR is awarded first only.
The call of out might be by the book, but I can't believe that's intentional.
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Paxil classaction

The runner was saying they didn't make an attempt towards second base. There was no contact between F3 and the runner.

That is why there is no call on a tag.

Irrelevant

I don't think an attempt is relevant. They didn't overrun the base, they fell off of it.
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Oliviya cam

Runner falls off the bag and are tagged, they are out. The obstruction of F3 didn't cause her to fall off the bag. She side stepped F3 touched the bag and fell off it.
Out in my opinion. Am I wrong.

Why did she fall off?
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GLASS PIPES

Really? Define "overrun"? Where does the book say the runner must run through the base? Where does it state a runner must remain on their feet? The runner can do cartwheels, somersaults and back flips after touching 1B if they want as long as the umpire doesn't judge she made an attempt to advance.

The ONLY way this runner is called out is if the runner literally stopped while in contact with the base and then, in a completely separate action unaffected and unrelated to those attaining 1B, the player loses contact with the base and then is tagged with the ball while off base.

Call it however you want, but the correct call is nothing without the runner making an attempt toward 2b.

Where does the book say that the runner has to literally stop on the base and fall off of it? Maybe it does say that. I'm just asking where.

I smell troooooooooooooollllllllllsssss

What is the average wing beat of an unladened swallow?

ive heard of unladened spitters.

Overrun means to run beyond or past. They don't have to stay on their feet but to be entitled to the overrunning exception they do have to run past the base. Once they do that I agree with you that they can do whatever they want as long as it isn't an attempt.

If they ran to 1st and stepped on it and fell forward, I'd go with what you're saying as well, though as written a runner tripping over 1st base is liable to be put out.
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How To Roll A Joint

being obstructed at first base negates any put out at first. but one could always try.