I've looked all over and have found multiple types of save files that all state that they're vanilla saves, and while they may be, I've found that when loading them the save count is upwards of 300 or so. So, I've gone through, removed every single mod of mine, even the DLC Hearthfire, Dragonborn and Dawnguard as well as the HighRes Texture Packs. The only esm's attached to this file are Skyrim.esm and Update.esm. This should allow far more people to use it right out of the gate, and if you're like me who gets annoyed by the save count being very large, Starts off as Save # 1

Edit: According to the Nexus themselves, this save file will work with the Special Edition of the game. (I personally haven't tried it yet) If there are any problems, let me know and I'll do my best to fix it.
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Edit 2: So apparently this was listed under Models and Textures for a while and I didn't even notice that. A fellow member pointed it out to me but by then it had already been changed by a mod. Also, for some reason, there was no file attached to this listing, so I've gone and already made a Vanilla save from SE, zipped it and uploaded it here, also a #1 save file. 

Thank you to theicedevil for pointing out the incorrect mod location.
Thank you to Zaldiir for fixing said mod location.