The official instrumental that I got from your link is only a 160 kb/s mp3 which means that it totally has that problem built into it to begin with, but you're definitely making it worse by exporting the project as an mp3 again (especially if you didn't change Audacity's default bitrate setting for mp3s which, IIRC, is just 128 kb/s) and I assume SoundCloud then makes it even worse.
About formats: If you were told that mp3s are better than wavs, then you got pranked. Wavs are basically raw sound waves, so they are perfect in quality but pretty large in filesize. Lossless compression formats like FLAC find ways to optimize the way in which the data is presented so that the quality stays the exact same, but the filesize is slightly smaller, and lossy compression formats like mp3 find ways to present a close-ish approximation of the data so that the filesize is greatly reduced but the quality is also reduced, and this becomes especially apparent in the high frequencies — mp3 has specifically been designed to preserve the quality of a human voice as much as possible, but cymbals are higher than that so they get compromised.
When it comes to the quality of a lossy format, there are two extra things to take into account: what is the bitrate, and how many times has it been re-encoded into a lossy format. The higher the bitrate, the less quality you lose, and with something like 320 kb/s, most people won't be bothered by the loss in quality and can only tell the difference when there's a direct comparison and a good listening environment, but less than 200 kb/s gets pretty annoying to listen in my opinion. And every time you open an mp3 in Audacity and export it back into mp3, some more quality is lost, because you're not getting an approximation of the original raw data anymore, you're getting an approximation of a previous approximation. And if you have a low quality mp3 and you open it in Audacity and export as a wav, the result sounds the same as that low quality mp3 but it has the large filesize again, so it's pretty inefficient either way, so you should avoid using lossy files as any kind of source audio for a project if possible. In this case, it doesn't look like it's possible, so the wav-sized file that only really has mp3-quality audio in it is the lesser evil.
Tape and analog saturation happens when you have a tape machine or analog processing equipment and your input volume is pretty high so the sound starts to distort, but in small amounts, that distortion actually sounds really pleasant. And then there are software plugins that attempt to digitally emulate that sound, and even some freeware plugins like the ones developed by Variety Of Sound can achieve a pretty cool sound these days. Normally you would use them just to add a bit of color and analog feel to your mix, but they do create some higher harmonics so they also work for trying to replace the low quality mp3 high end with something else.
...and yeah, the fact that you're using Audacity definitely doesn't help with any of this. Audacity is not trash per say, but it's an audio editor, which means that it's designed for things like removing noise from noisy recordings, and not for mixing. Mixing on Audacity is like being unable to draw at the same time as your eyes are open so you have to alternate between drawing and looking, and only being able to use the eraser on the most recent line you drew so you have to do the whole thing again if there's something you did earlier that you want to change later. Digital audio workstations (DAWs) such as Reaper and Cakewalk don't have these problems, so you can be adjusting the first thing in the plugin chain and hearing those adjustments going through the entire plugin chain in real time if you want to.
Hope this helps a bit.