No, Embassy will evaluate your IELTS separately from any institution. College will gain money if your study permit is successful, IRCC knows that and they will scrutinize your application. Your score is poor, but some people have gotten a visa with your score, it is all how you present your application and your SOP.
We generally advise against private colleges, as they don't have as much value as public colleges. We also suggest that you study and apply for IELTS again, as the IELTS marks show that you are not very proficient with the English language. This will be a problem in Canada, as English is the language of communication.