Ok, I did get it to work now.
Turns out I had two DEM folders, one in the Plan-G installation directory, and one under %userprofile%\documents\Plan-G Files.
Not sure if I accidentally created the first one, or if that was the one used by V2 of Plan-G, but V3 definitely needs the one under My Documents.
(To confuse things further in my case, the DEM folder beneath the installation directory did already contain an altitude tile, so I assumed that was the one used. But there was also one in My Documents, and that was the one that mattered.)

I did check the manual, but that sometimes refers to a "Plan-G\DEM" folder and other times to "Plan-G Files\DEM", so I went with the one I saw right there, in the installation folder...

I used to fly in North America, and the elevation profile always worked fine.
Now I was doing some flights in South America, downloaded and extracted the appropriate tile (J), but no profile is shown.
It looks like Plan-G does know something about the elevations, as the start and end-point ground elevations are indicated correctly, but the "zig-zag" graph is missing. Is that only available for certain areas in the world, or did something go wrong here?

Here's the content of my DEM folder (beneath the folder Plan-G3.exe resides in):


And this is what I get if I enter a flightplan for "SPLN UAN LAY SPTE":

This seems to work with some KML files, but not with all of them.
If I, for example, download a KML file from FS-Duenna (e.g. this one - download from options on the right), Plan-G seems to import it, but nothing is displayed.

It looks like I have to make the following changes, in order for it to be shown:

Replace
<kml xmlns="http://earth.google.com/kml/2.2">
with
<kml xmlns="http://www.opengis.net/kml/2.2">
   
And Change
<Style id="tlffStyle"><LineStyle><width>4</width></LineStyle><PolyStyle><color>7dff0000</color></PolyStyle></Style>
to
<Style id="tlffStyle"><LineStyle><width>4</width><color>7dff0000</color></LineStyle></Style>   

It would be nice if Plan-G could accept these types of files without them having to be edited manually.