dimensional analysis - How can the speed of light be a dimensionless constant?

This is a quote from the book A first course in general relativity by Schutz:

What we shall now do is adopt a new unit for time, the meter. One meter of time is the time it takes light to travel one meter. The speed of light in these units is

$$\begin{align*}\end{align*}$$

$$\begin{align*} c &= \frac{ \text{distance light travels in any given time interval}}{\text{the given time interval}}\\ &= \frac{ \text{1m}}{\text{the time it takes light to travel one meter}}\\ &= \frac{1m}{1m} = 1\\ \end{align*}$$

So if we consistently measure time in meters, then c is not merely 1, it is also dimensionless!

Either Schutz was on crack when he wrote this, or I'm a dope (highly likely) 'cos I can't get my head around this:

The space-time interval between different events at the same location measures time, and between different events at the same time measures space. So they're two completely different physial measurents: One is a time measurement using a clock, the other a space measurement using a ruler. In which case the units of $c$ should be $ms^{-1}$

Does Schutz correctly show how $c$ can be a dimensioness constant?