I was a math tutor for a couple of years in college. It is difficult, but this can be corrected (assuming the student is willing to put in the work).

I had gaps in my knowledge when I started tutoring as well. Factoring polynomials was always a struggle for me. I would pretty much guess and check until I got it right, and that was good enough for me to pass all of the tests in high school. Of course, until I got to multivariate calculus and I needed to factor more complicated polynomials as part of a much bigger problem. As you said, it compounded. Did I learn my lesson? No. I bought a calculator with a CAS to do that part for me. But when I started tutoring, I felt that it was unfair to my students to not know how to do something correctly, so I forced myself to learn it properly.