The disavow tool is technically a recommendation to Google to ignore the toxic links, it does not have to disavow any, let alone all of the sites that you suggest. The point of sorting out a penalty is is performing penance and proving you won't do it again. Taking the easiest step first without recording efforts that show that you have tried to resolve it manually won't make you pass the manual review.
I know this for a fact as the agency that got us in a penalty in the first place performed two disavows before I took over and lifted our penalty. I manually checked over 10,000 and recorded my efforts for all, only then was it lifted. I understand that you want to fix it quickly as it is your business, but there is no quick fix other than playing by goggles rules on this one and showing that you understand that spammy links are not good. I recommend email out reach and recording everything on a goggle doc to send through with your re inclusion request. Best of luck