Konrad1

From within the font...
Versions String: Version 1.024;Fontself Maker 3.5.0
The good thing about Fontself Maker is anybody can make a font.
The bad thing about Fontself Maker is anybody can make a font.
Multi-mapping is a bad practice that experienced font developers avoid.
There is a reason Adobe is against it and Google Fonts does not allow it.
You will not find any multi-mapped glyphs in Monotype's Helvetica Now.
Because support is spotty in OSs, text shapers, PDF libraries, and applications.
Given that the developer used Fontself Maker means they are on a Mac.
And that particular combination of OS and text shaper (Coretext) happened to work.
Doubtful they tested anywhere else.
I did not examine this closely enough to determine exactly what is causing Affinity problems - but my guess is the ridiculous level of multi-mapping, and including ranges, is confusing Affinity.
Regardless this font is junk.
I have a pet peeve against purposely broken "free" fonts.
Think it is a very sleazy practice.
All the characters with diacritics which were there are now mapped to just the base.
 
Be honest. Just make a limited version and let users know it is limited.
Don't mislead and confuse them by mapping all the characters they actually need for anything other than English to just the plain base character.
And the curly left/right single and double quotes are mapped to the plain versions.
I assume there were actual left/right versions.
Sleazy.
If you just gotta have this font, you can edit it and remove all the multi-mapping, and also remove the unneeded old control characters - and then it may work properly in Affinity. So if you only need Basic Latin uppercase, and some of the normal punctuation, that may be a good option.
Could also take a shot and buy the commercial version. May work.
 

@Konrad1  The problem is the Aquire font. Wadda mess.
Multi-mapping is when a glyph is mapped to more than one Unicode code point.
This can quite often cause problems for applications with those fonts.
Adobe strongly recommends against multi-mapping.
Google Fonts does not allow multi-mapping at all in their fonts.
It is a dumb idea to do it.
It appears this free version of the font was created by deleting the extended glyphs,
and then multi-mapping the deleted glyphs code points to the base glyph.
For example the capital A glyph has 14 code points mapped to it.
That is beyond stupid.
And it uses ranges of code points in addition to single code points.
Many applications will choke on that.
This "free" font is basically unusable.
It may work in some situations, but not cross-platform, and not in many applications.
Could not find the commercial version to take a look at it and see if it too is a mess,
but, regardless, I would not trust anything made by this "font designer."

Well, it's been logged as a bug (AFB-4986) in the thread you linked to so hopefully, it's being looked at with a view to finding a resolution.

So it looks like a Windows Font Rendering Issue in Publisher, opening your sample file on Mac I see this...

Hi @Konrad1 and welcome to the forums,
I'm not seeing that, for me, it's renders and exports like this... are you sure you don't have any tracking applied to the text...