Let $D$ be a region such that every loop/closed path is homotopic to the trivial loop in $D$.
This construction is given in Gamelin's Complex Analysis (2001):
Let $\gamma$ be a closed path in $D$ with starting/ending point $z_{0}$. Let $\Sigma(s,t)$ be a homotopy between $\gamma$ and some point $z_{1}$ such that $\Sigma(0,t) = \gamma(t)$, $\Sigma(\frac{1}{2},t) = z_{1}$, and $\Sigma(s,0) = \Sigma(s,1)$ for $s\in[0, \frac{1}{2})$. We will construct a function, $\Gamma$, forming a homotopy between $\gamma$ and $z_{0}$ in the following way: For $s\in[0, \frac{1}{2})$, put $$\Gamma(s,t) = \begin{cases} \Sigma(t,0) &\qquad 0 \leq t \leq s \\ \Sigma\left(s,\frac{t-s}{1-2s}\right) &\qquad s \leq t \leq 1 - s \\ \Sigma(1-t,0) &\qquad 1-s\leq t \leq 1 \end{cases} $$ and for $s\in[\frac{1}{2},1]$, put $$\Gamma(s,t) = \begin{cases} \Sigma(t,0) &\qquad 0 \leq t \leq 1 - s \\ \Sigma(1-s,0) &\qquad 1 - s \leq t \leq s \\ \Sigma(1-t,0) &\qquad s \leq t \leq 1 \end{cases} $$
$\Gamma$ seems to have all the desired properties of a fixed endpoint homotopy but Gamelin does not address what seems to be a problem point to me and what my professor agrees is the only problem: is $\Gamma$ continuous at $(\frac{1}{2}, \frac{1}{2})$? Here's a picture:
There would be no problem if it was the case that $\frac{t-s}{1-2s}$ became arbitrarily close to either $1$ or $0$ because $\Sigma(s,0) = \Sigma(s,1)$. But this isn't true. The only thing I can imagine is that maybe we can come up with some bound on the size of the loop as $s \to \frac{1}{2}$. Any ideas?
Because $\Sigma$ is continuous in $s$ and satisfies $\Sigma(\frac{1}{2}, t) = z_{1}$ (constant), for every $\varepsilon > 0$ there must exist a $\delta > 0 $ such that if $\frac{1}{2} - s < \delta$ then $$ |\Sigma(s, t) - z_{1}| < \varepsilon$$ for all $t$.