My question is easily stated: are all continuous finite-dimensional real representations of $\text{SL}_n(\mathbb{R})$ algebraic representations?

This is false if you drop the word "continuous" (e.g. using noncontinuous field automorphisms of $\mathbb{R}$).

It is also false if you work with $\text{GL}_n(\mathbb{R})$ instead of $\text{SL}_n(\mathbb{R})$. For instance, there are many examples in the answers to this question. However, all of the continuous examples there use the determinant in some way, so they don't give examples of non-algebraic representations of $\text{SL}_n(\mathbb{R})$.

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