In contrast to Saudi Arabias (or Qatars) military engagement in Syrias war there is quite comprehensible reason behind Hezbollahs involvement.
Hezbollah is subject to clear and present existential danger in case of the victory of mainly radical sectarian insurgents who for years have demonstrated their deadly anti – shia violence in Syria and Iraq.
There was and is obvious reason to assume that a CIA backed and Israeli supported „rebellion“ won’t stop at regime change in Syria but would also immediately cut the life line of Hezbollah in Lebanon. With the constant and increasing Israeli threat at the southern gates of Lebanon Hezbollah cannot afford to lose the Syrian supply channel.

Having shown why Hezbollah accepts significant casualties to ensure its survival it is barely comprehensible why Saudi Arabia goes at lengths to beat and remove the Syrian government and Hezbollah.
None of them poses a threat to Saudi Arabia, nor have they attacked Saudi Arabia. There is not even a frontier.
Essentially there is one neighbouring country that is highly pleased with Saudi Arabias huge support for the anti – Hezbollah forces: Israel, the country that not only occupies Palestinian land but also has attacked and fought every surrounding Arab country.