TL;DR version: anyone being put in jail deserves an explanation. None can be given when they are a non-violent drug user.
Drug addicts are a high risk group for triggering cost in social safety nets. The risk of driving under the influence of drugs are higher for drug addicts, there is a higher risk that parents fail their responsibility while under drugs, and as with other addictions there is an increase in risk for failing to pay rent and having enough money to pay for food. Drug addicts are also more likely to get sick and needing health care (relevant for countries where the government guaranties health care).
Since the government is expected to step in and pay for the social safety net, it then make sense that they implement laws to manage the risk and reduce costs. One way to do so is to make drugs illegal, especially those direct associated with risks that increase government costs.
If the government did not have any responsibility to supply a social safety net then there would be no need to have those laws.
If I just stabbed someone, there is a clear explanation for why I am being put in jail. If I am possessing a drug, why a I being put in jail? Because of some hypothetical possibility? Because I belong to a group of people that have a higher chance of doing something that might end up in something bad happening?
Justice needs to answer an individual why they are about to be deprived of most of their liberties. Putting someone in jail is the worst thing the government does to individuals; such a severe action requires a personal explanation.
One such law that exist in most countries triggers if you drive while under the influence just by yourself on a lone road. You can still end up in jail. In the eye of the legal system it is bad enough that an accident might happen. Same for driving a car deemed unsafe. There is similar laws around the world against guns and explosives, where simply possession is enough to run fault of the law.
As with most laws, first time offenders generally should not end up in jail (US can be a bit strange and unfair on that point). However a lot of laws all over the world are simply about what might end up happening given a risk factor.
The current drug laws are more analogous to putting someone in jail if they purchase alcohol and put it in their trunk because there's a chance that they might drink and drive.
Also, isn't imprisoning people a huge cost itself?
Alcohol is however generally not defined as a hard drug. Soft drugs would be alcohol, marijuana, sleeping pills and sedatives, while hard drugs includes heroin and cocaine. Netherlands has a very strict line between the two in both enforcement and punishment. Netherlands has a war on hard drugs, but is celebrated for not having one for soft drugs.
When it comes to alcohol and driving, Sweden for example uses a system that directly punish that circumstance. Driving licenses are very expensive to get and take a lot of time. Getting hit by a DUI not only mean you have to retake the tests and pay for a new license, but you are also likely to get a ban for up to 3 years. For many that is a very harsh punishment and one enough to discourage drunk driving. For sever DUI with blood values over 1.0 you also risk jail sentences up to 2 years. Any DUI involving other drugs than alcohol is regulated under the harsher rules. You do not need to trigger an accident for this to happen.
Drug laws does not need to be all or nothing. Scientists could/do define the risk profiles for different drugs in different circumstances and law makers can make more lenient or strict laws depending on those risk profiles. Sadly culture do play a role in permitting some high risk over other, but that doesn't mean risk management is bad explanation for why laws exist that punish non-violent drug user. It only make them inconsistent.
> Scientists could/do define the risk profiles for different drugs
The scientific risk profiles I've read have alcohol right up there on the top of the ranking for hardest, most damaging drugs. Sometimes it's at #1, but it's always right there in the big 4 with cocaine, heroin, and meth.
"Not defined as a hard drug" is a policy choice, almost entirely because of cultural/historic reasons, as you point out. Prohibition didn't work for alcohol, and it's not working for other drugs.
DUIs, on the other hand... harsh prosecution of DUIs is fine because they're crimes with victims, where other people are put into serious and immediate danger. That's a better approach than the overly broad, costly, and ineffective policy of simply incarcerating everyone for drug use.
It will hopefully never end up as bad that people go to jail for it, but as culture is changing and public opinion about smoking goes more negative, tobacco as an accept social drug are loosing acceptance. In theory they could change it so that smokers loose access to public health care but I doubt such solution is preferable over simply making smoking illegal.
Last year they even made an half-attempt at making smoking illegal during a very dry period under the argument that smoking was a major risk for causing fire.