Let $(B_t)_{t \geq 0}$ a Brownian Motion and $\xi_t := \sup \{s \leq t; B_s = 0\}$. Then we know (by arc-sine law, see René L. Schilling/Lothar Partzsch: "Brownian Motion - An Introduction to stochastic processes", Theorem 6.19)
$\mathbb{P}[\xi_t < s] = \frac{2}{\pi} \arcsin \sqrt{\frac{s}{t}}$
for all $s \leq t$. Thus
$\mathbb{P}[\exists t \in (0,\varepsilon): B_t = 0] = 1- \mathbb{P}[\forall t \in (0,\varepsilon): B_t \not= 0] = 1- \mathbb{P}[\xi_{\varepsilon} = 0]=1$
for all $\varepsilon>0$ which means that you can (a.s.) find for every $w \in \Omega$ a sequence $(t_n)_n$ such that $t_n \to 0$, $B(t_n,w)=0$.
And this means that $(X_t)_{t \geq 0}$ can't be a Brownian motion: Let $w \in \Omega$, then we have $X_{t}(w) \not= 0$ for all $0 (by definition of $T$). So there can't exist a sequence $(t_n)_{n \in \mathbb{N}}$ such that $t_n \to 0$, $X_{t_n}(w)=0$.