So my wife nagged me to install Spotify on my phone and test. And indeed, Spotify actually delivers music out of "the puck".

I am slowly starting to suspect that there is a bug in the software (ever heard that before) where whatever protocol is used for streaming music via Bluetooth is not working...

So after fighting with this for several weeks now I have some conclusions to draw.

The first and probably to no ones surprise:
The software from B&O is absolutely catastrophic!
Many moons ago I was teaching software programming and there is no question that the responsible for the Core would receive an F.
In the close to 30 years I have worked in the software business I have seriously never seen anything so bad released to the general public.
Ever.
Not even close.

In a small defense of B&O there is plenty of software, at least on the Windows 10 platform, that is behaving really badly in a distributed environment.
But that explains far from all the problems B&O has with their system.

As already indicated the app is total shait when it comes to detecting the hardware.
I have installed both a Core and a 35 at the same time so I can compare the diffs.
I finally managed to get things going by connecting a LAN cable to both boxes.
Windows was then able to detect the boxes and after connecting using Media Player to each box it seemed like it provoked an update of the firmware that then allowed me to find the boxes in the app.

I am connecting using Bluetooth and all involved software has a really hard time keeping track of what should be having access to the boxes.
This does however not free B&O from responsibility because most problems lie there.

When the Core detects a new device it will happily drop whatever connection it currently maintains to connect to the new device.
The Core can play perfectly well for hours and then all of a sudden start dropping the current connection.
After that it maintains a new connection for only a few minutes.
If you then check the app (assuming that you can actually connect to the box) it will sometimes all of a sudden have switched to radio.

The list of problems with the Core, and to some extent the 35, is equally staggering as endless.
One example,
I have configured both Windows 10 and the boxes to not reconnect automatically.
Does not matter.
On startup of the laptop, it connects to both boxes, often killing whatever is already playing.

If the Core is what B&O considers the future then let me predict the future:
Bankruptcy and all that is worth anything will be bought by a Chinese company.
Tragic end to a classic brand but when there is a board that can only be described as utterly clueless then there is nothing else to expect.

By the way, I managed to figure out the issue regarding the initial question.

It is a combination of software hogging the speaker resource and bad handling of volume control.

Firstly, just because you decide that a certain source for sound should be what should be sent to the speaker does not mean that it actually will be.
For instance, pausing a YouTube video might (will) hold on to the Bluetooth output and to switch to another source (say VLC) you have to close the browser to release the Bluetooth output.

Also the volume is handled, let's call it, extremely individually.
This means that just because the volume when playing from one device was good, it does not mean that the volume level for the next source is even possible hear.
A drastic increase in the volume might be needed.

In other words, a lot of work is needed from a lot of programmers to get a distributed system working in a way which is user friendly.