Different frequencies, and different intents. The dwemer sought to make themselves all gods as the Numidium, and tried to hone in on the correct frequency the heart emitted, while being invaded. The Tribunal had time to research the heart, the tones, and their intent.
I would agree with your first sentence, but not your second.
I think it's a little too dismissive to assume that "the dwarves killed themselves because Kagrenac was too proud to bother letting an intern check his math." It's not like he came up with this plan overnight, this was the long-term goal of their
entire society and philosophy, and what their whole history as a people were leading up to doing.
If there was any flaw in their actions, it was in the whole of their philosophy to start with, but there is, again, no evidence that they actually failed at anything, and actually achieved their own form of transcendence (or should I use a different term... "Unnascendence?") The Dwemer are, functionally speaking, the Thalmor, but actually intelligent enough to know what they were doing, and having a proper chance of achieving it. They just didn't take the rest of the world with them.
If anyone did too little research, it's the Tribunal. They either set their sights too low or did think things far enough through, and only achieved a demi-godhood. For all Vivec's desperate posturing as some sort of philosopher king to gain some fragment of respect, he didn't even achieve as much as Tiber Septim did.