Of the three satellite geodetic techniques contributing to the International Terrestrial Reference Frame (ITRF), Satellite Laser Ranging (SLR) is generally held to provide the most reliable time series of geocentre coordinates and exclusively defines the ITRF origin. Traditionally, only observations to the two LAser GEOdynamics Satellite (LAGEOS) and Etalon pairs of satellites have been used for the definition of the ITRF origin. Previous simulation studies using evenly sampled LAGEOS-like data have shown that only the Z component of geocentre motion suffers minor collinearity issues, which may explain its lower quality compared to the equatorial components. Using collinearity diagnosis, this study provides insight into the actual capability of SLR to sense geocentre motion using the existing geographically unbalanced ground network and real observations to eight spherical geodetic satellites. We find that, under certain parameterisations, observations to the low Earth orbiters (LEOs) Starlette, Stella, Ajisai and LAser RElativity Satellite are able to improve the observability of the geocentre coordinates in multi-satellite solutions compared to LAGEOS-only solutions. The higher sensitivity of LEOs to geocentre motion and the larger number of observations are primarily responsible for the improved observability. Errors in the modelling of Starlette, Stella and Ajisai orbits may contaminate the geocentre motion estimates, but do not disprove the intrinsic strength of LEO tracking data. The sporadically observed Etalon satellites fail to make a significant beneficial contribution to the observability of the geocentre coordinates derived via the network shift approach and can be safely omitted from SLR data analyses for TRF determination.