The most well known double-bird-strike incident is probably the one where Sully landed a plane on the Hudson River.
https://en.wikipedia.org/wiki/US_Airways_Flight_1549#Takeoff...
-edit: ok, everyone had the same thought, haha -
There was this very famous double bird strike https://en.m.wikipedia.org/wiki/US_Airways_Flight_1549
There certainly have been bird strikes that disabled all engines.
What? It happens multiple times a year. They made a movie about a famous incident (US Airways Flight 1549). There's even events with four engine strike (Eastern Air Lines Flight 375).
So far this year, it's happened at least twice resulting in engine failure.
Jan - Air New Zealand NZ207 - aborted take off following dual bird strikes and both engines damaged.
~~Mar - Ryanair FR 4102 - dual bird strikes on landing resulting in injuries and hull loss.~~
Before that, in December 2024, Jeju Air Flight 2216 crashed on landing following dual bird strikes, resulting in the loss of all PAX.
These aren't as rare as you believe they are.
Selection bias
The lower the barrier to entry of the subject matter the lower quality the people discussing it. This crap is like the Kardashians for white nerds with stem degrees.
The people with the requisite dozen brain cells to common sense realize these problems are complex and keep their mouths shut.
But in the video of the plane taking off and crashing, there's no clear, obvious, or tell-tale "poof" of bird turning into exhaust as there often is in bird strikes.
That doesn't rule it out.
Constructing solutions for multiple-mode failures like this is a bad engineering smell. Almost always the solution isn't actually helping anything, and often makes things worse in whatever metric you're looking at. In the example here, having four engines makes the chances of total thrust loss lower, but it doubles the chance of a single engine failure. And the literature is filled with incidents of theoretically-survivable single engine failures that led to hull loss as a proximate cause (generally by confusing or panicking the crew).