This is true in general, in a way.
Consider a regex $r$. You can construct a DFA (or NFA) for the matching problem. Denote this NFA by $A$.
To solve the searching problem, you can proceed in several ways.
First, you can just solve the matching problem for the regex $\Sigma^*\cdot r\cdot \Sigma^*$.
Alternatively, you can construct an equivalent NFA for the latter regex. The form of such an NFA would be an initial state that guesses when $r$ starts (at which point there is a transition to the initial states of $A$), and every accepting state of $A$ moves to an accepting sink.
This NFA can be thought of as a generalization of the situation you describe. Indeed, when you determinize it, the equivalent DFA essentially tracks $r$, and whenever it fails the DFA "resets". This is not exact, since it strongly depends on $r$.
When $A$ is a chain, then indeed determinizing the NFA yields a DFA that follows the chain, and whenever there is a "mistake" (that is, $r$ is not met), it resets to the initial state. However, it could be that resetting might not go as far as the initial state. For example, when searching for $aba$, if you see $aa$, then the second $a$ should not reset all the way back, but rather to the second state, since the second $a$ may be the start of the regex's occurrence.
