I don't find "inflammation" to be a very meaningful answer because "inflammation" is a range of very complex biological processes.

We use the same word to describe the process by which you feel pain after a muscle injury but also the effects of asthma or processes that cause damage to your heart in heart disease but there are distinctly different paths involved. Systemic corticosteroids suppress most kinds of inflammation powerfully and can save your life but they will destroy you when used chronically.

I had that CRP test the day I after I got a contusion on my upper leg that had me off my feet with inflammation but the CRP does not detect that kind of inflammation because I got a low score like I have every time I've had a CRP draw when I was or wasn't having symptoms that could be related to CRP. There is certainly some inflammation that contributes to CRP but other kinds that do not.

There are at least 14 kinds of serotonin receptor on many cells in your body, in your gut, on your platelets, on your heart valves, immune cells, etc. Most people who try a few different SSRIs can usually find one that agrees with them but many people have some uncomfortable physical symptoms with their first SSRI that may cause off-target effects: because of those receptors on platelets, SSRIs act as blood thinners

https://pubmed.ncbi.nlm.nih.gov/17506225/

which could save you from a clotting event but kill you through a bleeding event.

If SSRIs work as advertised and raise serotonin in the junction then some will diffuse out into the fluid partition and have some effect on other cells so between that and off-target effects some immune cells must have be affected somewhat and there is some connection with "inflammation"

I am fascinated in pharmacology and biology how these systems have structure and how you can interrupt a process at multiple points, for instance there is a process by which a protein is made in the liver, cleaved by another protein in your lung (the ACE-2 receptor that COVID-19 attacks), which is regulated by a protein from your kidney that binds to a receptor. There are a few very successful drugs for high blood pressure and related conditions that block this system but there are several other hormone systems of a similar structure related to this that are not so well developed or understood. The targets of these drugs are all proteins that are coded for by genes so there is a straight line from the pharmacology to generic susceptibility to illness

Some mental illness, such as schizophrenia

https://pubmed.ncbi.nlm.nih.gov/27968732/

are heritable. This is the complexity of what we're getting in to when we draw a line from some drug that is known to bind to certain proteins to the change in your experience and behavior comes from taking a pill. A lot gets lost when we simplify it to the kind of pithy narrative one would expect from a salesman or politician.

I saw my nurse practitioner when I expected my stress level at work to explode in the near future and she offered me an SSRI to blunt my physiological reaction to stress. That plus fanatical amounts of exercise got me through a few challenging years of developing my own business. I felt more in control when I was under stress and never get flustered or mean to people and that the SSRI was helping me be a better parent.

I eventually got back into W-2 work and after I settled in, my primary core doc asks me if I want to stop the SSRI and I say sure. It has been a long strange trip since, I did find that parts of me were suppressed by the SSRI. One day I went into a rage because the room I was in was terribly cluttered and clocked 40,000 steps on the pedometer without leaving the house, cleaning it. I have been through waves of growth and consolidation and am definitely seeing meaningful things to do with my time.