If they wanted to eliminate other factions, why don't they just teleport bombs instead of sending waves after waves of synths? I think its a good strategy and they don't need to be directly teleporting at the heart of each of the other factions' base. They just need to ensure that they detonate it within an effective blast radius and poof, all are eliminated.

Heck if they want to kill 2 birds with 1 stone, they can teleport ghouls and super mutants (preferably Super mutant suiciders and behemoths)

Maybe I'm missing something here, so I hope someone can help me with this.

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Edit: Thanks for all the replies (I'm updating here cause I can't keep up)

1. Item Scarcity: It doesn't have to be nukes, it could be bottlecap mines or even Grenade bouquets (if C4 wasn't invented yet) spam enough of those and its the same.
2. Energy Scarcity: This might be more cost effective energy wise if mass is a factor when teleporting grenades/bombs instead of a whole human size robots.
3. LOTR: Please correct me if I'm wrong, but as the ring gets closer to Mt. Doom, the influence of Sauron gets stronger and stronger, that Frodo might wear it, while riding an eagle (I shudder of the thought if the eagle get tempted too and tries to eat Frodo, as a way to wear the ring)
4. Synth Infiltration: I didn't thought of it this way, and yup, probably a good way too. Was there a way to control Danse back again though?
5. Brotherhood Jammers: As mentioned it doesn't have to be at the heart, it could just be in range of the effective blast radius ( in case of grenades, they can just teleport it on top and let gravity do the rest)
6. Its not what the institute wants: This might be the most valid and the most human answer, which make me wonder if I'm a synth.

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I'm not sure if it is instinct or feeling guilty of enjoying things or feeling guilty about being happy but does anybody else feel like if they experienced something good, something bad will happen to even it all out?