Short answer, no, they don't.
Longer answer, it's complicated. There are, in essence, five different measures for the centre of a galaxy cluster, based on different physical properties of the cluster, and occasionally not in agreement with each other.
- BCG, the brightest cluster galaxy. It should sink to the bottom of the gravitational well, but that's only the case if the cluster had enough time to evolve. Of course, galaxy clusters are, cosmologically speaking, quite young objects and still in the process of forming, so this is not a given.
- Mean position of the cluster galaxies. Works for spherically symmetrical clusters with lots of galaxies. Problems: Poisson noise due to limited number of galaxies, identification problems (is this galaxy really a part of the cluster), asymmetry.
- X-ray: the hot intra-cluster gas emits X-ray radiation that we can observe - more and hotter gas means more and harder X-rays and if the gas is in equilibrium, the highest intensity radiation comes from the centre. Unfortunately, the gas is typically not in equilibrium, there are cooling flows, streams, cold cores and other things that make the measurement less straight-forward.
- (Weak) gravitational lensing. Measures the total mass of the cluster and doesn't care about ugly gas physics. Problems: Shape noise (measurements are based on mean ellipticity of background galaxies), resolution / smoothing issues with the mass map created from averaging over lots of background galaxies.
- Sunyaev-Zel'dovich effect. Again the hot cluster gas, but this time we have inverse Compton scattering of photons in the cosmic microwave background on the electrons in the gas. The problems with gas physics should apply here, too.
So, there are other methods to find the centre of a galaxy cluster. This paper suggests that ususally the weak lensing centre is offset with respect to both X-ray and BCG. I am not aware of anyone using the mean position of the galaxies as a measure for the cluster centre.
This paper evaluates the cluster centre offset between measurements in X-ray and the SZ. This is interesting because they both trace the hot intra-cluster gas.