Yes, something is going wrong.
Normally, in a midi file, each track (which actually means nothing to MIDI, it's just a extra compartment in a midi file) is assigned to a single midi channel, then each channel (and the track involved) is assigned to a specific instrument/sound/drum kit/whatever. ALSO, the device that is receiving the midi (sound module, VST, etc) needs to be in POLY mode) so that it recognises the different channels.
Then, all the sound will appear, the group will sound like a group, the orchestra like an orchestra, etc..
But this does not happen of it's own accord, someone has to do something to set it all that way.
Midi files normally has that side of things set. It's rare that this is NOT set in the file. Whoever created the midi file probably went to a LOT of trouble to get it to sound correct. They would have noticed if it did NOT sound right.
Also, most multi-timbral sound devices default to polyphonic, rather than OMNI.
Are you sure that you haven't done something, maybe accidentally, to change something. Maybe the playback device/VST has got set to OMNI, maybe the channels have got cancelled from the midi data, or the Program Change instructions (that assign the different instruments, have got cancelled (or they might still be there but are being disregarded)?
One thing you could do is to take one of the midi files that is playing wrong, put it into a .ZIP file, and attach it to your next message. Then someone here can check it and make sure that the file is correct. If the file IS OK, then it's something else, but we're a step nearer sorting the problem out.
MIDI files always play fine for me!
REgarding your last para, there's still a problem when you play via Reaper? Do you need to add in the Channel assignments, or the specific instruments, or BOTH, to get it to come right? Those settings probably WERE there in the file originally, how have you lost them? What have you done with the files BEFORE you tried them in Reaper?
Maybe the midi files use certain Channels, but your sound device uses other channels, and you have to mess with settings to get any sound at all and maybe end up in OMNI? Or alter the midi file in some other way.
Nice would be a midi file before you do anything, then another copy when you find that something is wrong but just before you load it into Reaper.