How LIGO, LISA, etc. Detect Gravitational Waves
The point of instruments like LIGO and LISA is to measure time-varying changes in the distance within different arms of the instrument. In the case of an arm oriented in the direction of an incoming gravitational wave (GW), the length of the arm will increase and decrease, while an arm oriented perpendicular will remain unchanged.
The method for measuring this is to split a laser beam, send each half down the different arms, and then recombine them and look at the interference pattern. If the arms have identical lengths, then the beams will destructively interfere. If, however, one of the arms becomes longer (or shorter), the recombined beams will be out of phase, and you will no longer have perfect destructive interference.
The key to being able to measure this sort of thing accurately is two things:
Arm length, so the beam traveling down the altered-length path has time
to accumulate a large enough phase shift for the interference pattern to
change enough to be measured. Longer arms lead to larger phase shifts,
which is why LISA aims to have arms that are 2.5 million km in length.
The wavelength of the light: the shorter the wavelength, the smaller the
actual shift in arm length needs to be to make the beam shift appreciably
in phase. If the recombined beams differ in phase by 0.001%, then the change
in the interference pattern will be very hard to detect and measure. But if they differ
in phase by 50%, then the difference will be obvious. Shorter wavelengths
make for larger phase shifts.
If you want think of the interferometric measurement as a kind of timing measurement (that is, we are, in a sense, measuring the difference in time it takes light to travel down the two arms), then optical light is vastly better than radio for "timing" measurements.
As you pointed out, there are projects that use pulsars to try to detect gravitational waves (e.g., pulsar timing arrays).
These work by measuring time delays in the arrival of radio pulses from millisecond pulsars. However, it's important to note the limitations: in order to measure the arrival time accurately, you have to observe a pulsar over a period of several minutes (or more) and add up all the pulse arrivals to get a single, accurate value. (You are not measuring changes between the arrival of one single pulse and the next.) You then repeat this measurement several weeks or months later to get another arrival time. Since you're taking measurements weeks or months apart, you can only detect variations on that same time scale, which is why pulsar timing arrays are hoping to detect GWs with periods of months to years (e.g., from binary supermassive black holes). The kinds of GWs that LIGO detects -- with periods of fractions of a second -- are completely beyond what pulsar timing arrays can do.
We cannot use optical light from stars, because we know of no star that produces the exquisitely regular periodicity of a millisecond pulsar.
As a final comment, your scenario of photons that "arrive regularly at 15 photons every microsecond onto a CCD" has the problem that Poisson noise (a.k.a. "shot noise") means that you wouldn't actually get 15 photons every microsecond; you'd get around 15 every microsecond (a typical sequence might look like this: 16, 15, 18, 15, 14, 14, 15, 23, 16, 12, ...). With that much variation, it's really hard to detect subtle variations in intrinsic signal strength.