In the laboratory notes it is indicated that it is ill advised to eat the solid formed from this experiment. An esterification reaction, in which salicylic acid is treated with acetic anhydride.
If it is for the acid added as catalyst either sulfuric acid or phosphoric acid. Why isn't it possible to neutralize it with a strong basic?
The addition of water should be enough to quench (destroy) any remaining acetic anhydride and turn it into acetic acid, and this latter one is more soluble than aspirin so it should be possible to crystallize aspirin without any acid.
The question appears in this lab notes http://www2.xula.edu/chemistry/crs-orgleclab/Organic_Lab_1_2230L_Web_Files/23_Aspirin_notes.pdf
I think it is self-evident that your basement shouldn't become a pharmacy. I think it has more to do with the acids used in the synthesis. Does the industrial process differ from the synthesis presented in the article with regards to this aspect (acid catalyst and their neutralization after the process)?