Yes. There is probably a quick proof using ultraproducts but I am not sure. Here is a very rough sketch of a proof.

$1\le p<2$ case. Let $\varepsilon_k\searrow 0$, and suppose $(x^n_i)_{i=1}^n\in X$ are $(1+\varepsilon_k)$-equivalent to the unit vector basis of $\ell_p^n$'s. We will use a theorem of Dor that functions that are equivalent to $\ell_p^n$'s are essentially disjoint. Inductively, pick an infinite sequence $(y_i)$ which is equivalent to the unit vector basis of $\ell_p$ as follows. Let $y_1=x^1_1$. By Dor's result for all $n$, $(x^n_i)$ are essentially disjoint, say they live on disjoint sets $I^n_i$'s. For a large $n_1$, $y_1$ restricted to some (in fact, many)) $I^{n_1}_i$ has small norm. Pick $y_2=x^{n_1}_i$. Then find a large $n_2$ so that both $y_1$ and $y_2$ have smaller norm on some $I^{n_2}_i$, and continue this process. We also want our choices be an unconditional sequence. So a bit more care is needed to pick $y_i$'s incorporating gliding hump argument so that they are equivalent to a block basis of the Haar basis: Suppose $y_1$ is finitely supported. Pick a very large $n_1$ so that still a very large subset of $(x^{n^1}_i)$ have essentially the same coordinates on the support of $y_1$. Then take $y_2$ be a difference of two while also ensuring the process above. Then the resulting sequence $(y_i)$ is unconditional and they are large on disjoint subsets, then you can use a square function estimate to show the upper-$p$ estimate. The lower-$p$ estimate is automatic since $p<2$.

For $p>2$, you can use Kadec-Pelczynski's argument to get a sequence of peak functions. For any $\delta>0$, consider the set
$M(\delta)=\{x\in L_p: m(\{t:|x(t)|>\delta\})\ge \delta\}$. If a sequence is unconditional and belong to some $M(\delta)$, then it is isomorphic to $\ell_2$. Using this we can pick a sequence $(y_i)$ of peak functions equivalent to a block basis of the Haar basis as above from the collection $(x^n_i)_{i=1}^n\in X$ which are $(1+\varepsilon_k)$-equivalent to the unit vector basis of $\ell_p^n$'s.

*Dor, Leonard E.*, **On projections in (L_1)**, Ann. Math. (2) 102, 463-474 (1975). ZBL0314.46027.