$\newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\epsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\Sigma} \newcommand{\thh}{\theta} \newcommand{\R}{\mathbb{R}} \newcommand{\F}{\mathcal{F}} \newcommand{\E}{\operatorname{\mathsf E}} \newcommand{\PP}{\operatorname{\mathsf P}} \newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}}$
For $i=0,\dots,\lfloor\sqrt{t}\rfloor-1$, we have $t-i=ct$; here and in what follows $c$ denotes various positive quantities (which possibly differ from one another even in the same formula) bounded away from $0$ and $\infty$ over all real $t\ge1$. So, the sum in question is \begin{equation} s_T:=\sum_{t=1}^T (1 - (ct)^{-\alpha})^{c\sqrt{t}} =\sum_{t=1}^T \exp\{-ct^{1/2-\al}\}. \end{equation} If $\al\ge1/2$, then $\exp\{-ct^{1/2-\al}\}=c$ and hence $s_T$ is $cT$ and not $o(T)$. If $\al<1/2$, then $\exp\{-ct^{1/2-\al}\}=o(1)$ as $t\to\infty$ and hence $s_T=o(T)$ as $T\to\infty$; in this case, we even have $s_T=O(1)$.