Let's say we have a general point process, where the point in time events are some kinds of failures. The mean time between failures is given by $t$. By definition, the average failure rate of the point process becomes:
$$\lambda = \frac{1}{t}$$
Also, if we observe the point process for a certain interval, $u$ and expect the average number of events falling into this interval to be $N(u)$, the failure rate should also be:
$$\lambda = \frac{\Bbb E (N(u))}{u}$$
Equating the two expressions above, we get:
$$\Bbb E (N(u)) = \frac{u}{t} \tag{1}$$
And it is easy to show that this is the case for a Poisson process. Blackwell's theorem also suggests this is true for a general renewal process, but only if its non-arithmetic (if we want it to hold for any $u$) and we go "well into its lifetime".
For a general point process (ex: a renewal process), we observe the events falling into the interval of size $u$ and want a single estimate of the failure rate within that interval. One line of reasoning goes that since you're asking for a single estimator of the failure rate, you're essentially saying you want to fit the best Poisson process to the data (since the Poisson process is the only process that has a constant failure rate). And since the best estimate of $\Bbb E (N(u))$ is simply the number of events, $n$ observed falling into the interval, the best estimate of $\lambda$ should be:
$$\lambda = \frac{n}{u}$$
And this implies that the best estimator for the mean time between failures should be:
$$t = \frac{u}{n}$$
The claim is that this should hold for any point process, not just Poisson point processes. Are there any holes in this argument?