Should I give up on a math PhD if my first abstract algebra class was a C+? I did it never having written a proof ever.

[sampelton52]

In my freshman year of college, I had absolutely no math background nor any interest in mathematics. However, I started reading about logic (in the context of philosophy), and got very interested in it. I impulsively enrolled in my university's introductory abstract algebra and intro to proofs classes. While I got an A on the latter, I ended up with a C+ on the former because I very badly failed the first exam (I had never written a proof in my life). In the sequel class, however, I improved to a B+, and I have As in graduate courses on representation theory (one in the math dept, one in the physics dept), type theory and category theory, and algebraic coding theory, as well as As on most other pure math courses I've taken since then (I had taken linear algebra and gotten A before that, and also have As in undergrad analysis, graduate mathematical logic, and graduate computational complexity). I am taking graduate courses in algebra, analysis, high-dimensional probability and complex analysis the fall of applying, and have pretty good relationships with the instructors for each course (the algebra prof taught my rep theory course which was small, and he liked me), and I'll have about 7-8 graduate courses at the time of applying across math, CS and physics.

I think I have pretty strong research experience, and my letters from my mentors will be good. At the time of applying, I'll have two preprints, one of which (on the relationship between set theory and type theory wrt. the constructive analogous of large cardinals) will be submitted to a journal in logic within a few weeks (and hopefully approved -- my advisor, who is a prominent set theorist, seems to think it will be). I just wonder if the C+ in Abstract Algebra I means I should give up on getting into a PhD program at all, I am afraid it'll have me autorejected at every program. I do really want to study mathematics at the graduate level (ironically either algebra or logic), and I have done a lot of research that shows I can, but I am afraid.

For context, I go to a ~T30ish small university (LAC-like) in the US. I'm an international student. I am studying for and planning to take the math GRE subject test, and I think I can score 850+ because I tend to be a good test taker. Should I apply this cycle or would I be wasting money?