The money has to come from somewhere. Tuition doesn't cover the cost of operating those European public universities. Most of the cost is borne by taxpayers. More students would thus mean higher taxes.

To an extent this also applies in the USA. Public universities in most states have relatively low tuition (although still higher than their European equivalents) for in-state students. But they are unable to meet market demand, so many students take out loans to pay tuition at lower-tier private universities.