The Rwandan government, or at least individuals involved with the Rwandan government at the highest levels, were making the broadcasts.

If anything they would have cracked down on opposing broadcasts by claiming that, since Tutsis are the historically privileged beneficiaries of colonialism who were collectively responsible for the violent oppression of Hutus, any pro-Tutsi or anti-anti-Tutsi sentiment was unconscionable hate speech while any pro-Hutu or anti-Tutsi bigotry was not technically racism. The power to ban “hate speech” is the power to define “hate speech” and hateful people will use against you the very same weapons you propose to fight them with.

Not really. This is widely believed but attempts to rigorously study that end up finding no evidence for it.

Interesting that you picked Rwanda. The go-to example for the dangers of free speech is normally the Nazis. For instance I'm thinking of a study that attempted to correlate growth in support for Hitler with availability of his speechs on radio or via rallies. It found IIRC no correlation at all.

In fact Hitler was regularly censored, deplatformed and so on yet it didn't stop him, only perhaps slow him down a bit. On the other hand he spent vast efforts on suppressing, censoring, vote stuffing and de-platforming in various nasty ways anyone who spoke out against him - all managed entirely privately via his various organisations, whilst the weak Weimar Republic were unable to stop him.

Dictators are invariably big fans of suppressing free speech, because they know censorship works in favour of whoever is in power. The reason democracy and free speech are so tightly connected is because in a democracy the people in power aren't meant to be able to do anything to stay in power beyond winning the approval of their citizens.