The Ukranian embassy in Tehran already have a statement on their website saying it's an engine malfunction and are ruling out a terrorist/rocket attack (reported by CNN and NYT).

Right, but the question is whether that's a reasonable conclusion for them to have made already.

The pilots of the plane would be in constant contact with the nearest airport during and right after take-off, especially when there are technical problems.

It's not hard to imagine the conversation would involve like:-

PILOT: One of the engines has failed catastrophically, we need to make an emergency landing, can you clear the runway back in Tehran or is there another airport we can go to?

But such a statement wouldn't rule out anything. Catastrophic engine failures can be caused by missiles.

It would take everyone a few days to determine the cause, especially the people spreading unverified informations such as 'it was shot down by a missile'

It wouldn't take very long for the relevant parties to know whether they accidentally shot a missile at a 737. For a mechanical issue, there would be much more uncertainty (although it's possible the pilot reported the issue to ATC).

And even then, "ATC, we have a catastrophic engine failure" from the pilots doesn't discount that the catastrophic engine failure was caused by a missile. The pilots don't magically get to know what caused the engine to blow up.

An independent investigation is impossible in the current environment. You either choose to believe Iran or not.