The cops have absolutely no legal right to bring you to a psych ward unless you are
1) an imminent danger to yourself or others
2) "gravely disabled" to the point of not being able to provide food or shelter for yourself (in the US).
Ive had a few welfare checks. They ask if you're OK etc. Tell them you're fine & you're done.... Period!


Legal rights exist only to the extent that they can be enforced.
It's very unlikely a police officer has ever gotten in trouble for bringing a person to a psych ward after a psychologist has demanded a welfare check and speaking with a person.
Cops are allowed to use their subjective judgment of whether you are an imminent danger to yourself or others. If they they think you are lying, or don't believe you, they can bring you in. Even if they don't think you are lying, it's really impossible to prove their subjective judgments are somehow complete bullshit.
An important thing to remember is that in order for them to make that determination during a welfare check, they must have access to you (depending on the country). If you don't answer the door, don't respond to what they say, and possibly are not there, it's very unlikely they have the ability to break down the door without a warrant unless someone has said you are about to ctb (depending on the country), and even then, it's very iffy because if a cop is breaking a door it involves property damage and so they are unlikely to do that unless they have clear reasons to need to do that.
cops and mental health professionals are also not likely to spend an entire day waiting for you to come out if you don't respond. however, they may pretend to leave and wait outside the door for an hour, because sometimes people getting a welfare check will say nothing and then once they think cops/mental health workers are not there, they will simply immediately try to leave and then the cops/mental health workers are outside and suddenly, it's no longer possible to say nothing to them (because not being able to communicate may indicate you are gravely disabled or mentally disturbed).
in-patient involuntary care can be extraordinarily expensive, degrading, and often leaves people more suicidal then when they came in. Often time in-patient care makes society feel better (see, we tried to do something!) and makes doctors and nurses richer (they get paid for highly specialized care that consists of you sleeping in a locked area on a cot while they doom scroll on their phones). in-patient care in a country without universal healthcare can be well over $10,000 and possibly even much more than that. if you are in a country with universal care, you can check the rules of the nearby in-patient facilities. if they aren't that bad, in-patient care may not be the worst thing ever if it's free, although it can effect your rights and the loss of autonomy can be degrading
if you are sad, and have been sad for years, and are older, then you may not want in-patient care no matter what since that sort of extreme depression has a lower recovery rate. if you are young and having a first depressive episode, people often recover from those and in-patient care may make sense. good luck and sorry you are here.