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Your observation RHS $\in$ LHS is correct. LHS is not a unique function but a set of functions differing by a constant.

I take issue with your usage of $C$. There is not really a meaningful definition of $C$ for an arbitrary antiderivative. For example, what is the constant term of $\cos(x) + 3$? You could say it's 3 but what if we rewrite it as $u(x) + 4$ where $u(x) = \cos(x) + 1$?

Note that if $g \in \int f$ then $\int_{t_0}^t f = g(t) - g(t_0)$. Whatever constant is included in $g$ cancels in $g(t) - g(t_0)$ so $g$ and $t_0$ don't have anything to do with each other.

If instead we write $F = \int_{t_0}^t f$ then we have $F(t_0) = \int_{t_0}^{t_0} t = 0$. This equation forces a certain constant term of $F$. A different choice of $t_0$ can (but doesn't necessarily) lead to a different constant term of $F$.