The singularity is not inevitable.
From General Relativity, we know that the more an entity goes down a gravitational well, the larger the time dilation effects it experiences. Then, there's no need for a new mechanism to prevent the singularity, since gravity is its own show stopper, as matter don't flow in without consequences.
It’s true there's no force pushing out, but there's no need for it. Even if matter can't get out or affect the outside universe in any way, its time intervals get larger and larger as it falls down towards the core, so it can't get quite there either.
We reach a situation similar to the "Achilles and the tortoise" paradox: Matter would need an infinite amount of time to reach the singularity, so it'll never develop completely, no matter the unthinkable amount of time the black hole appears to last from the outside.
Just like matter can't reach the speed of light exactly due to the infinite energy this will require, matter falling into a black hole can't form a singularity, due to the infinite time this will require.
The mechanism that prevents the collapse into a singularity is, precisely, the same that initially removes energy and matter from the outside world into the black hole.
As you said, there’s a place (the event horizon) where infalling matter stops influencing matter outside... but this event horizon is subjective. Each entity or observer perceives a different event horizon, depending on its location in the gravitational field of the black hole, and this is also true for each layer within the black hole, no matter how deep we go.
The core of a black hole reaches a condition where each inner layer stops communicating with outer ones, and as the black hole gets more dense thanks to the infalling energy, the inner layers keep infinitely falling towards their "final" locations, without ever reaching them exactly. Each layer experiences it stops communicating with the outer layers, but as the inner layers also experience the same thing, the net effect is that each layer experiences a big rip as time dilates, no matter where the other layers are located within the black hole’s potential well.
The universe can get older by trillions of years, while mere seconds pass for the entities travelling towards "the singularity", until black hole evaporation reaches them from the outside.
A black hole is like a huge energy waterfall that takes eons to fall, but evaporates from the top before reaching the bottom, and this process can take from 10⁵⁰ to 10¹⁰⁰ years to observers outside the black hole (depending on its mass), while the same thing lasts between microseconds to days for the entities and processes trapped within.
Black holes are energy storage vaults that use gravity to lock energy by “compressing” it in ever increasing time spans. So in this sense, we could say black holes are really “bigger on the inside”.
