No.
Imagine a guy gets into a horrible car accident and breaks a leg and some ribs and had blood coming out of his face. People see the accident and watch him drag himself home. No one calls 911. No one tries to help him or even attempts to assess if something is wrong. They just assume he is okay and that he can splint his own leg, cure his own punctured internal organs, and wipe the blood off his face without wondering what’s causing the bleeding or if he's even capable of doing any of these things. Obviously this is stupid, and in this situation we would call 911 and be morally outraged by a bystander who refuses to call 911 because they think this guy should fix his own broken organs and body parts. For some reason a broken leg is seen as “injury that should be attended to by a professional” but mental health issues are not. Not only do many people believe that mental health isn't a part of physical health that needs to be attended to in the same manner but also that professionals need not be involved. Maybe this has something to do with how many shootings we have in this country? (One large mass shooting at the time I’m writing this, sadly more will have happened by the time I post this).
I have a rotator cuff injury. At no point in time has anyone ever snorted at me and said “why don’t you pull yourself up by your own bootstraps?” It doesn’t matter if the problem is genetic (I can rotate my arms further than most people) or created by something I did (I lift weights a few times a week.) here’s a problem- it hurts and it won’t stop hurting and I need to see a professional.
I think writing can be very therapeutic but it should never take the place of seeing an actual professional (as in "I'm going to journal out all my feelings and figure this shit out on my own.") I don’t know why people snort at the idea of a therapist. Look at the statistics about mental health and come talk to me. US soldiers are more likely to die of suicide than they are to die in combat. At least once a year a young man walks into a crowded area and commits a mass killing. People unable to cope with something turn to substance abuse or self destructive behavior or to abusing others.
For some reason, psychologist and therapists are viewed with more derision than are MDs. Because “anyone can just listen to someone talk.” 1) "Listening to you talk" isn't what therapists do, 2) seriously, no one listens to you fucking talk anyway, so what are you complaining about? No, really, being serious. Many people literally do not have someone who will listen to them. Or there are people who ostensibly listen, but really, we know they are not: the are judging, screaming at us, using the information against us, etc. So yes, you do need someone to listen in the particular way a professional would, but that isn't all they do. (If what you think they do is so simple, please apply to a MSW, PsyD, or Clinical PhD program, get in, succeed in it, do your internship, then come talk to me about how simple it is.)
But I went to a therapist once and they sucked! Therefore therapists don't work! Why would you think therapists would be any different from medical doctors, handymen, or chefs? Some are going to work for you, and some aren't. One does not represent all. In the case of therapists, the type of chemistry you have with that practitioner is a lot more important than for say, a dentist. You need someone you can trust. You need someone who is capable of seeing what you're going through, and calling you on your shit. You need an outside perspective, because sometimes you can't even see the hole you are digging for yourself, and sometimes some part of you is really, really committed to digging that hole as deep as possible.
Where does writing fit in? Journaling, or writing through your feelings, or whatever, is fine, but will never take the place of seeking professional help if you really need it. Writing can be a tool you use, but much like I can't write away my rotator cuff injury, you can't write away your depression.
Where writing can be really useful is if you're someone who has a hard time articulating their feelings. Things can get lost in the chain between heart and words: sometimes people are not even aware that they are feeling something at all. This can happen when someone has emotional numbness. There's something under the numbness, but it hasn't come up to the surface. Or they might be aware that they are feeling something, but not what it is. Some people experience depression by feeling so dejected they cannot leave the house--others who are depressed can still go to work and go about their daily lives, but still also be depressed. One of my anxious friends has described the feeling of anxiety to me as feeling so stressed about social interactions that they are avoidant of them. I sometimes have an adverse reaction to caffeine that makes me extremely anxious but it looks really different: I still go and do whatever it is I have to do but I feel an intense panic, like I might go crazy at any second. Not everyone would think to label that emotion as "anxiety."
Where writing is really useful is when someone doesn't do a lot of emotion regulation, or isn't really self-aware or articulate about their emotions. Not everyone will realize that the bad day of work made them yell at their spouse, but writing it down and looking back at it might help you see the connection. Not everyone will see that a pattern in their behavior is occurring until they see it written down over and over in their own handwriting. Articulating emotions is hard, particularly if you are not articulate in general, or are not in touch with your feelings, or are not a very verbal person at all. It is a skill set, and one that writing may help, particularly if you find it easier to work out exactly what it is you think by writing, as opposed to just talking or thinking to yourself. Writing can help you see patterns and help you work out what you actually think or feel about something, which can be useful for treatment. Cognitive Behavioral Therapy has a number of practical exercises you can do in written form, like challenging negative cognitions. But if you need treatment, don't diagnose yourself and just assume "if I sit down and think hard enough, I can solve this on my own."