Count the run if INT
 

I am a softball umpire, which might be why I have this question about baseball.

One out, bases loaded, ground ball to shortstop. F6 steps on second for out #2;
then attempts to throw to first. The runner from first interferes with the throw and the INT is called.
Umpires convened.

Is baseball the same as softball, meaning the runner closest to home would be out?

In this case, in softball the runner from second to third would be out #3,
because the runner from third crossed the plate before the INT.
Why would the run not count?

The first question would be what did R1 do to interfere? Continuing to advance and continuing to engage in a bona fide slide would not be interference. So we are left with an illegal slide or intentional swipe at the throw while in the basepath or abandoning his normal baserunning and somehow interfering with the throw while peeling off. We would call the B-R out and no run would score.

Why would the run not count?

No run scores if the third out is made by a runner being forced to advance or by the BR before reaching 1st base safely.

Wouldn't the 3rd out be a force out, even if it is due to interference? If the lead runner is out, then no, but if it's the runner from 1st, then I believe so.

The INT above was hands-raised, possibly protecting face. It was interference, rules cites not needed.
BUT, it was a retired runner, already out by the play at second (F6 steps on second for out #2;).

In softball, interference by a retired runner is penalized by the active runner closest to home being out. The trailing runner, such as the BR is out of the INT is by an active runner.

Hence, my question about why the run did not count as the runner from third reached home before the INT. The runner closest to home at that point would be the runner from second.

I don't do softball, but this sounds rediculous. It seems to give the offending team an advantage. In (HS) baseball, this would be a Force Play Slide Rule (FPSR) violation and an inning ending DP and no runs would score.
For any other INT by a retired player, the umps would have to determine who the D would have made a play on and that runner would be out. If they couldn't decide, the runner closest to HP would be out.

I don't see how calling the runner closest to home out is and advantage to the offense. The mind reading part (in red) is unnecessary and hard to defend.

Maybe I misread the play. It seems in your situation a play was made on the BR. Therefore he would be out and no run would score due to being the third out before reaching first base safely.