Basically, the WoSD 'was' (still is?) a mechanism by which authoritarians could impose control on society by oppressing society's undesirables. Echos of the Nazis, engender a police state, control Black people, etc etc.
But the WoSD hasn't gone so well, so it (as the main phenomena) was been supplanted by the War On Terrorism (cue the Patriot Act...) That has gotten a bit stale (but is still in effect?), so now it's ... Russians? Covid-19 and all the relevant restrictions?
Or the GOP has decided to stop being indirect and now is going blatantly full out towards overt authoritarianism using The Big Lie as the mechanism? In some parallel universe, liberals are now being rounded up and put into FEMA re-education camps on President-For-Life Trump's orders because they tried to steal the election (and every future one as well).
There is no one Single True Motivation to this sort of social phenomena - they are all different expressions of the sociopaths in charge's drives for power.
This is a good book that shows some of the motivation for a War on Drugs is not in the name of health, safety, security... but the exact opposite. The War on Drugs sorta failed in these aims, so now "terrorists" are the new excuse for a 'soft' police state.
As commenters here, and Rat Park , and "Drug Warriors and Their Prey"  explain, such a simplistic notion about the causes is not correct, along with the assumptions about the government's true motivations. There are more complex reasons, based in psychology and sociology, that explain why people have drives to alter their consciousness and why government policy is not actually what it seems a lot of the time.
This metaphor of a "war" is deliberately misleading, but simplistic enough to be jammed into people's heads - it is the 20th century equivalent of "Reefer Madness". Thus it is no surprise the Right is so fond of this, even now... they are obsessed with trying to mold society into what they want, versus trying to actually understand and solve problems appropriately. Thus their stereotypical dismissal of 'science' and all that...
But as I heard recently, conservatives have always been on the wrong side of history. US society is now evolving and legal marijuana is never going to go away now - primarily because of the money involved (and that is another essay)