Yemen has had an interesting although rather underappreciated role in international affairs over the centuries. As you say, control of Suez and the Red Sea is important. I don’t have the impression that SA is doing much in Yemen with ground troops. Is that true? Engaging in Syria would certainly give their military a different kind of experience than just bombing cities in Yemen.
From reading Simon Winchester’s book Krakatoa, I could see the subtle influence Yemen had in Islamic affairs. After the Haj to Mecca, Indonesian Muslims went to Yemen where training and preparation was made for rousing Muslims in Indonesia to overthrow the Dutch because the volcano showed that “God” no longer supported them. I wonder how many other Muslim related international situations have had a genesis or an assist from Muslim forces residing in Yemen. More info would be good. If, in fact, agitators have been traveling from the Haj in SA to receive training and advice in Yemen, this would certainly help explain their interest in controlling Yemen.
I get the same impression about Saudi ground forces. I sometimes joke to myself when I read about Saudi military intervention and wonder if their armed forces aren’t made up of Philippine and Pakistani migrants. Perhaps its just a stereotype but the stereotype does exist that Saudis live a subsidized lifestyle and the backbone work in the country is done by immigrants.