> My understanding is that in GR, massive observers move along timelike curves $x^{\mu}(\lambda)$

... where $x^{\mu} : \mathcal E_O \to \mathbb R^n$ appears to be some coordinate assignment to the set of events $\mathcal E_O$ in which the observer under consideration ($O$) took part

(perhaps not even "just any" coordinate assignment, but rather an assignment which represents the events $\mathcal E_O$ in the order in which observer $O$ took part, as a "path $\mathbb R^n$");

and $\lambda : \text{(subset of) } \mathbb R \to \text{(subset of) } \mathbb R^n $ is a real-valued parametrization of those coordinates

(perhaps not even "just any" parametrization, but again rather a parametrization which represents the order of observer $O$'s indications monotonously) ...

> and if an observer moves from point $x^{\mu}(\lambda_a)$ to $x^{\mu}(\lambda_b)$

... or focussing on the physical: if observer $O$ moves from one particular event (which we happened to denote by coordinates $x^{\mu}(\lambda_a)$) to another particular event (denoted by $x^{\mu}(\lambda_b)$) ...

> then his clock will measure that an amount of time $t_{ba}$ given by the curve's arc length;

This seems to indicate a misunderstanding, either of terminology and conventions of notation, or even more profoundly:

in the theory of relativity, "time" means

- foremost an indication of an observer

(such as the indication $O_A$ of observer $O$ of having been in coincidence with observer $A$, having jointly taken part in coincidence event $\varepsilon_{A O}$),

- and in a derived sense a (real) number value $t$ assigned as a clock reading to an indication of an observer (or also assigned to the entire event, of which the observer indicated its participance), such as any function $$ t : \mathcal E_O \to \mathbb R$$

(or perhaps not even "just any" such function, but rather only such functions which are monotonous wrt. the order in which observer $O$ took part in the events of set $\mathcal E_O$).

Obviously, there is a tremendously large number of distinct such clock reading functions $t$;

any two of which are not necessarily continuous wrt. each other, much less differentiable, or even smooth (differentiable to any order), or even affine (proportional to each other).

On the other hand, the arc length of the timelike curve of a particular observer (between two particular events, such as between the two events $\varepsilon_{A O}$ and $\varepsilon_{B O}$ in which observer $O$ took part, or between two particular indications of the observer, such as correspondingly between the two indications $O_A$ and $O_B$ of observer $O$),

a.k.a. the duration of observer $O$ between these two indications,

a.k.a. the proper time of observer $O$ between these two indications

(as far as the term "proper" is permissible at all, since it suggests the possibility of "improper" notions as well),

is usually denoted by the letter $\tau$;

for the above case explicitly as $\tau O[ \, \_A, \_B \, ]$.

Ratios of durations are unambiguos real numbers, i.e. quantities to be measured.

Further, the readings $t$ assigned to indications of a given clock provide a measure of its coresponding durations $\tau$ as far as for any three distinct indications $O_H, O_J, O_K$ holds:

$$ (t[ \, O_K \, ] - t[ \, O_H \, ]) \, \tau O[ \, \_H, \_J \, ] = (t[ \, O_J \, ] - t[ \, O_H \, ]) \, \tau O[ \, \_H, \_K \, ].$$

If (and only if) this was satisfied, the clock under consideration (incl. the clock reading assignment $t$) is said to have been "good", or having "run evenly".

> Why is this so?

As a matter of definition;

in particular of the notions "duration $\tau$ (of an observer, between two of its indications)", and of whether a given clock was "good" or to which extent it was not.