Comment to the post (v2):

1. The statement that

Work $$W[\vec{\gamma}]~:=~\int_{\vec{\gamma}} \vec{F}(\vec{\gamma})\cdot \mathrm{d}\vec{\gamma}$$ does not depend on the curve $\vec{\gamma}:[t_i,t_f]\to\mathbb{R}^3$ from $\vec{\gamma}(t_i)=\vec{r}_i$ to $\vec{\gamma}(t_f)=\vec{r}_f$

is a non-trivial statement about the force field $\vec{r}\mapsto \vec{F}(\vec{r})$. In fact, it is (or is equivalent to) the conventional definition of a conservative force.

2. The statement that

$W[\vec{\gamma}]+W[\vec{\gamma}^{-1}] = 0$, where $\vec{\gamma}^{-1}$ denotes the reversed curve

is a triviality for a force field $\vec{r}\mapsto \vec{F}(\vec{r})$.

The core of OP's question (v2) seems to be resolved by the following fact:

1. On one hand, the statement

Work along any closed loop is zero

is a non-trivial statement. In fact, it is (or is equivalent to) the conventional definition of a conservative force.

2. On the other hand, the statement

Work along a closed path is zero whenever the closed path just retraces the same curve segment twice, back and forth in opposite directions

is a trivial statement true for any force vector field (without explicit time dependence).