Something must have gone wrong with my reasoning.
UPDATE: This is the answer to my question provided by Dr. Ben.
Where is $|\theta|<1$ used in recursive method derivation of invertibility of MA(1)?
Given a zero-mean AR(1) process $\{X_t\}_{t \in T}$:
$X_t = \phi X_{t-1} + \varepsilon_{t}$; $\{\varepsilon_t\}_{t \in T}$ is $WN(0, \sigma^2)$
There are infinitely many solutions satisfying the recursive equation above, and there is only one stationary solution which is $X_t = \Big[\sum_{i =0}^{+\infty} \phi^{i}L^{i}\Big]\varepsilon_t$.
Assume that $|\phi| < 1$, and $\{X_t\}_{t \in T}$ is constructed to be a NON-STATIONARY solution of the recursive equation above.
However, I found it contradictory that,
$X_t = \phi X_{t-1} + \varepsilon_{t} \iff X_t - \phi X_{t-1} = \varepsilon_{t} \iff (1 - \phi L)X_t = \varepsilon_{t}$
As $|\phi| < 1$, $1 - \phi L$ is invertible, and $(1 - \phi L)^{-1} = \sum_{i =0}^{+\infty} \phi^{i}L^{i}$.
Therefore, $X_t = \Big[\sum_{i =0}^{+\infty} \phi^{i}L^{i}\Big]\varepsilon_t$, implying that $X_t$ is STATIONARY.
Is this a contradiction?