Archaea and Eubacteria have different Ribosomes. In the world of biology, if DNA is the source code then ribosomes are the compiler.
In other words, while both families use DNA to store information and RNA to translate it, all the rest of the machinery used by cells in that process is different. This is why there are no intermediates. Likely (and this is a guess, if an educated one) this split happened before cells were even "cells". That is, whatever the pre-cellular replicons were that eventually led to life, there might have been many variations on (what eventually became) the ribosome, but two won out and gave rise to two distinct lineages of cellular life.
So, if I'm understanding correctly, these pre-cellular replicons are distinctly less efficient at survival than both archaea and eubacteria, hence their not existing in any (recognizable) form today. Is this correct?
This opposed to the idea that the archaea existed, then the bridge (which went extinct), then archaea.
I should start by saying that we don't know, for sure, that pre-cellular replicons don't still exist. They may, and we may just not have figured out the right place to look for them.
That said, gaining a membrane is a huge advantage for life. The current working hypothesis is that life started as a series of autocatalytic chemical reactions, but for this to work there must have been just the right combination of chemicals in just the right conditions all at the same time. You can imagine that it wouldn't take much of a stray current to wash all that away, and you're back to square one. By surrounding themselves with a membrane, pre-cellular replicons could begin controlling their chemical environment, enriching it with the raw materials needed and regulating the conditions.
Again, though, just because cellular life would be expected to be more robust and, therefore, more successful than acellular replicons, you wouldn't expect those acellular reactions to immediately halt once cellular life arose.
If I had to guess, I'd say that acellular replicons may have persisted for quite some time, but ultimately succumbed to the Great Oxygenation Event (https://en.wikipedia.org/wiki/Great_Oxygenation_Event). Shortly after cellular life arose (or possibly beginning a bit before cells came on the scene), life cause a massive shift in the Earth's chemistry, freeing vast quantities of oxygen and changing what had been a reducing environment to one that is, on the balance, oxygenating.
Oxygen is, to put it bluntly, nasty stuff. It tends to break down complex molecules very readily, and so any acellular replicons that may have survived up to the Great Oxygenation Event were likely wiped out soon after.
Like others said, we don't really know, but most likely, they just got eaten, ingested, or sopped up by cellular life when it arrived. They couldn't have been much more than symbiotic molecules, not complex things like modern cells with organelles that we know today.
In other words, while both families use DNA to store information and RNA to translate it, all the rest of the machinery used by cells in that process is different. This is why there are no intermediates. Likely (and this is a guess, if an educated one) this split happened before cells were even "cells". That is, whatever the pre-cellular replicons were that eventually led to life, there might have been many variations on (what eventually became) the ribosome, but two won out and gave rise to two distinct lineages of cellular life.
The rest, as they say, is history...