Breached Dome feels like a replacement for Shock!. Both are simple "runner hits it, it fires" pain providers that fire from everywhere, including archives. Breached Dome has no trash cost instead of the 2 of Shock! but Shock! rarely got installed anyway as runners are typically wary of running on unrezzed cards in Jinteki remotes.
The main differences are that Breached Dome does meat damage rather than net damage, which makes it a bit weaker since runners are more likely to have meat damage protection since it's more often a direct kill threat, and that Breached Dome additionally trashes the top card of the stack.
Trashing one card of the stack is not directly a problem, like the meat damage it's typical Jinteki death-by-a-thousand-papercuts. But when there are twp Breached Domes in archives a hit on archives suddenly means 2 cards from your grip and two that would soon enter your grip going into the heap. That's four cards that, things like Paperclip excepted, won't be useful to you anymore without recursion tricks. This is quite a discouragement. In particular Industrial Genomics: Growing Solutions decks will profit from the runner running archives less.
Because of the two influence Breached Dome is reasonably splashable. The best out-of-faction identity I see working with it is Skorpios Defense Systems: Persuasive Power since Breached Dome allows Skorpios' ability to fire more often on the runner's turn.
With regard to power level I'm not 100% convinced Breached Dome is very strong. It's good, it's absolutely a contender for many Jinteki decks, but it doesn't feel overpowered. It won't win you the game, it'll just slow the runner down. A card like Snare! on the other hand can win you the game when followed up with Scorched Earth. Granted, that requires a dumb or careless runner (looks away in shame).
This card works miracles for my shell game. Too many cards in hand? Discard a Breached Dome. Not enough ICE to protect archives? Discard a Breached Dome. Gotta throw away some throwaway agendas? Discard a Breached Dome.— snatch22