The origin of the spins of stellar-mass black holes is still controversial, and angular momentum transport inside massive stars is one of the main sources of uncertainty. Here, we apply hierarchical Bayesian inference to derive constraints on spin models from the 59 most confident binary black hole merger events in the third gravitational-wave transient catalogue (GWTC-3). We consider up to five parameters: chirp mass, mass ratio, redshift, effective spin, and precessing spin. For model selection, we use a set of binary population synthesis simulations spanning drastically different assumptions for black hole spins and natal kicks. In particular, our spin models range from maximal to minimal efficiency of angular momentum transport in stars. We find that, if we include the precessing spin parameter into our analysis, models predicting only vanishingly small spins are in tension with GWTC-3 data. On the other hand, models in which most spins are vanishingly small, but that also include a sub-population of tidally spun-up black holes are a good match to the data. Our results show that the precessing spin parameter has a crucial impact on model selection.