I think the problem here is how trains with impossible routes due to incompatible/unpowered railtypes handle this. Navigating as far as possible and then displaying "waiting for path" isn't helpful, but neither is reversing on what is likely a one-way track. It would be better if it never left the previous station and displayed a warning (both as a news item and on the train's status bar), "no route to next station" or similar. I'm sure there are other solutions, too.

If I forgot to electrify a piece of track, the electric trains should go to the last electrified tile, then turn around as if the non-electrified track didn't exist.

This is exactly what will happen when you use regular (block or pre) signals. For path signaling, this behavior is correct and changing this could cause more problems than good. By the way, if the trains go on, you may never notice where the problem is. If the trains stop at the station, away from the problem, it won't be any easier to locate either.
In such a situation, at best, a warning could appear about the inability to reach the destination due to the lack of appropriate tracks (not only the lack of traction can be a problem here).

By the way, if the trains go on, you may never notice where the problem is.

Yes, but it would be more consistent with e.g. a missing track piece hidden under a bridge.
You would then be notified "train X made -5,000 € last year" and would closely follow the train en route to where it surprisingly changes direction and then find the missing track or missing electrification.

Here however - I mean the "steps to reproduce" from the OP are really dumbed down - think big: On a big busy multi-track crossing with two dozen switches and path signals securing each of the entries: You would think the train "is waiting for free path" because the junction is so extremely busy, not because the necessary track isn't build yet.

For path signaling, this behavior is correct and changing this could cause more problems than good.

I have no idea of how the path signal algorithm works, but I imagine something like this:

const paths_possible_list = PathFinding.collect_all_possible_paths( my_position, destination );
bool could_reserve_track = false;
for each possible_path in paths_possible_list
{
         if ( PathSignalBlockingMe.try_to_reserve_path( possible_path ) )
         {
                  could_reserve_track = true;
                  break;
         }
}
if ( !could_reserve_track ) this.ShowWarning( "Waiting for free path." );

If so, one could just check for paths_possible_list.length < 1 and show a different error message.

Something like

"No path to destination could be found."

or

"Destination not reachable."

I think in general OpenTTD could serious benefit from "Destination unreachable" message. I have a very similar issue with boats that cannot get to their dock ... especially when using rivers, good luck figuring out it got stuck going up-river.

I really hate that by default it just starts to run around blindly hoping that some day magic happens and it can find its route. But without telling me, the human, about it, that is in many cases rather unlikely to happen.

For trains with PBS it seems it is slightly "worse", as instead of running around blindly, it just gets stuck with a weird (read: unrelated) message .. yeah, that is confusing to any (new) player :)

Currently what you get in most cases is the popup that either "ship is lost" or "made last year", but both are not visible on the vehicle itself, which is also confusing. And also shows up very very late.

So yeah, lets make this a bit more user-friendly, one way or the other :) Shouldn't be "that hard" (famous last words!) :D

I think in general OpenTTD could serious benefit from "Destination unreachable" message.

That already exists; if a train is unable to get to its next destination for any reason, the pathfinder will flag the train as lost and this is displayed to the user as "Cannot reach [destination], X km/h" on the status bar, with a warning icon visible to the left and in the train listing.

Idea: If a train has a destination and can't pathfind to it, occasionally (e.g. once a month) try to invoke the pathfinder in first a way that ignores railtypes, and if that succeeds give the player a specific warning that the train can't find its destination because of incompatible track.
I don't know how hard it would be to make the pathfinder callable like that.