Rrerku

An observer in general relativity is defined as a future directed timelike worldline
beginalign*
gamma:I subset mathbb R &to M \
lambda &mapsto gamma(lambda)
endalign*
together with an orthonormal basis $e_a(lambda) in T_gamma(lambda)M$ where $e_0(lambda)= v_gamma, gamma(lambda)$ and
beginalign
g_gamma(lambda)(e_a(lambda),e_b(lambda))=eta_ab~. qquad (1)
endalign
Here, $v_gamma, gamma(lambda)$ is the velocity of the worldline $gamma$ at the point $gamma(lambda)in M$ and $g$ is the metric tensor field on $M$. The time measured by the clock carried by this observer between events $lambda_0, lambda_1$ is defined as
beginalign
tau_gamma = int_lambda_0^lambda_1 dlambda sqrtg_gamma(lambda)(v_gamma, gamma(lambda),v_gamma, gamma(lambda))~.
endalign
However,
beginalign
g_gamma(lambda)(v_gamma, gamma(lambda),v_gamma, gamma(lambda)) = g_gamma(lambda)(e_0(lambda),e_0(lambda))=1 qquad (2)
endalign
which follows from the requirement of eq.(1). We are using signature $(+,-,-,-)$.

This is all standard definition. Suppose, we have another observer $delta$:
beginalign*
delta:I subset mathbb R &to M \
lambda &mapsto delta(lambda)
endalign*
and the time measured by his clock between the same two events $lambda_0, lambda_1$ is
beginalign
tau_delta = int_lambda_0^lambda_1 dlambda sqrtg_delta(lambda)(v_delta, delta(lambda),v_delta, delta(lambda))~.
endalign
From equations (1) and (2), we get $tau_gamma = tau_delta$ and this will be true for all observers measuring time between $lambda_0, lambda_1$.

However, I know that my conclusion is wrong. Can you point out where I went astray?

Your conclusion is correct, because what you are doing by saying that $g(v_gamma,gamma(lambda),v_gamma,gamma(lambda)) = 1$ is that the parameter $lambda$ is exactly equal to proper time. You can have different parametrizations $tildelambda$ of the curve $gamma$ that have $g(v_gamma,gamma(tildelambda),v_gamma,gamma(tildelambda)) neq 1$ and then, of course, they do not correspond to proper time of the observer on the curve.

Your conclusion from the OP just states that if you have two curves parametrized by proper time, then when they are evolved for the same amount of proper time, the same amount of proper time passes on them. A quite tautological statement!