Exactly as you say, but an icc color profile different from the color profile in use in the document, it always comes, in any case converted from a source profile to a destination profile, instead the exact procedure is that it must always keep the numeric values and never convert to destination (unless appropriate user requests).
In practice, with "no profile embedded" images it behaves well in this way.
Now that I remember well "the Adobe theory" there are 3 criteria of choice in color management in Adobe Suite.
2. maintains the embedded profiles
3. converts into a work profile
the right choice is 2, it keeps the numerical values embedded in the color profile of the file. Put simply, consider that photo as if it had no color profile applied and read only the numerical values applied to it.
Then in the PDF / jpg export phase etc, you decide to "keep the numerical values" and apply a color profile of your choice, which can be either the work document or other.
Practically if I have a fogra39 profile document and I insert images with the fogra39 profile embedded, the numerical values are read in the right way. If I change color profile to the working document and I put fogra27 example and I click Assign, the color profile in fogra27 will change and the numerical values will remain the same as before (and it works well), but if I then have a fogra27 document and insert an image again with different profile eg fogra39, this image will be converted and numeric values will change.
In practice, when inserting a photo AD always applies a color conversion in the destination profile !!!! Even if the Convert colors box is not checked in the color preferences.
I don't think the color assignment command works well.
The color profile of the document should be ASSIGNED to each image inserted, imported, linked to the document so as not to VARY the numerical values even if you work with different embedded color profiles.
Working appropriately with color management in the press, web, etc. (but above all in the press) is FUNDAMENTAL!
Incorrect conversion of numerical values can lead to serious financial damage during printing!
A trivial example if a 100% black is converted to a rich black 80-80-30-80 is not the same thing!
Not to mention color change in images or dominant etc ..
this thing is very delicate Attention Affinity !!!
It took me years of study to understand how the "good" thing works in Adobe, now here I find myself headed and above all with the awareness of a wrong management by the software.
Affinity we would love to leave Illustrator (because it's shit!) But you have to be able to do it without problems!