The memorylessness of the state, not of time. It is what makes the algebra tractable.
The hard part is choosing states, not solving equations. For "expected flips to see HTH", the useful states describe progress toward the pattern, not the flip count. Get that right and the rest is three linear equations.
Expected hitting times: E_i = 1 + sum over j of P(i to j) E_j, with the target set to zero.
A useful shortcut: for a random walk on a graph, the stationary probability of a vertex is proportional to its degree.
Diagnostic. If your equations come out inconsistent, the state is underspecified.