No, Saudi Arabia should not have sole say over the Hajj. The Saudi Arabian government have been involved in numerous human rights scandals in recent years that detract from them being adequate stewards of the Islamic faith. In 2020, this human rights record led to the rejection of a Saudi-backed consortium's purchasing Newcastle United football club on moral grounds, should such a country really be in charge of a religious pilgrimage if they are ethically unsuitable to be in charge of a football club? Islam needs a multinational body that organises the Hajj, the larger islamic world should run the Hajj not just Saudi Arabia.
Saudi Arabia have also sponsored war crimes against Muslims in Yemen, and have a terrible human rights record, offering virtually no protection for women's or children's rights, with children occasionally ending up in the country's opaque criminal justice system to be tried as an adult. The Saudi government have also been instrumental in repressing dissidents, human rights activists, independent clerics, and journalists, sometime going as far as murder. For such a violent and repressive regime to have control over Islam's most holy site is not fitting with the peaceful message of the muslim faith. Some people have gone further and advocated for all holy places to be open to all people, of all religious backgrounds, as they should be seen as belonging to humanity, keeping them as exclusively for people of one religion just helps sustain resentment and conflict.