Could Control Weather stop natural disasters?

No, and Yes

For the initial question

So does this mean that the hurricane/tornado/blizzard/sandstorm would only be temporarily stopped according to RAW?

the answer is "no", but not for any particular rules-based reason. The operative reason is that D&D 5e doesn't simulate natural disasters, or even non-disastrous weather, in much detail. There aren't inherently low- and high-pressure weather systems, atmospheric circulations of heat, etc. If there is a tornado or hurricane in the game, it's because the DM announced one, not because it emerged from a deep, emergent combination of underlying factors.

So if you magically downgrade a mundane tornado to a modest breeze (which the spell allows, all else being equal), you've stopped 100% of what the tornado, itself, is in the game. There just aren't ambient meteorological details present in the game to "restart" it later. Whether or not the normal weather reasserts itself is just a choice the DM makes, not something a player or PC can independently determine. For RAW considerations it is, at most, more a question of if the DM considers the storm's presence to be the "normal weather" or a place or not.

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For the second portion of the question

could a casting of Control Weather in the path of the storm (before it hits you, standing in the eye of a hurricane, etc.) have an effect?

the answer is an explicit "yes". Changing weather conditions like wind speed and precipitation are what the spell does. The only things it does. If a hurricane subjects a spellcaster to strong wind and driving rain, the spell explicitly can reduce those to less intense levels. Whether or not a particular character can apply the spell to do so successfully is a completely different matter-- the spell itself can produce the desired effects, and possible issues with the caster are separate.

I submit that an adventurer capable of casting an 8th level spell can, at least potentially, keep the spell going during violent storms. A 15th level adventurer is already superhuman, after all, and likely has access to resources that could support their ability to maintain concentration during weather-changing efforts. There are any number of reasons the effort might not work in a given situation, but nothing which suggests that it cannot work under any circumstances.

Whether or not the 5-mile radius of the spell is sufficient to completely break up a larger weather pattern is, as above, not defined well enough in game-mechanical terms to answer. Maybe there are decisive changes that could be made at high-leverage points in storm systems that could unravel those systems. Or maybe there aren't, but magic patches over the gaps anyways.