Last I checked, there aren't any Xeons with E-cores. However, you're right: the most direct competitor to Bergamo seems to be Sierra Forest, a chip that will have nothing but E-cores (and no AVX-512).
Even if they are not branded Xeon, but Atom, there already are a lot of Intel server CPUs with E-cores, e.g. Denverton Refresh, Snow Ridge, Parker Ridge and the recently launched Arizona Beach.
Now imagine Intel had that product and canceled it in 2017, and you will be living in reality: https://en.wikipedia.org/wiki/Xeon_Phi
Not only would something like 1024 cores with AVX512 be competitive with GPUs it would have the added advantage of being waaay more versatile and easier to program.
And I'm not sure it'll be that much easier to program - "few large cores" and "GPU waves" are now pretty well supported in the stack with mature tooling - trying to insert a new stack between the two would likely be pretty difficult as it doesn't really fit well with either established paradigm so likely needs something new to show it's benefits.
It was a big experiment, and it failed. Some of the brightest people have tried to push it to the limit. In my circle, I have never heard anyone missing Xeon Phi when it is discontinued.
The cores were in order. If you've never written code for in order cores, you may not appreciate exactly how heavily you're leaning on out of order to save your bacon. Things your used to just working suddenly don't.
The cores had a 2D layout, but Intel refused to tell you what it was and in practice it was impossible to optimize for. You ended up with variable (as in different on every node) latency for memory access that made programming again painful (and add this to the fact the cores are in order, so you really can't work around the latency).
Then there was the software stack that was just odd in some ways. I was researching I/O throughout, and things that just work on any other CPU (on any modern ISA) just didn't behave in a sane way.
So yeah, I was not at all sad when Intel cancelled it.
A chip with hundreds or thousands of E-cores should be ideal for any cloud provider to offer vCPUs with a high markup. For example, a company like Hetzner sells 1vCPU services for around 4€/month. Even if they don't overprovision and allocate 1 core per 1vCPU, a single chip can potentially earn them 500€/month, which leads to a break even point at around 2 years.