My personal experience is that it's true hence I would caution against too much rest. Wrist pain is a very vague term, no idea what you have, but I have battled RSI for over half a decade (mostly of the fingers) and at some point it got so bad that I wanted to quit my degree and it felt like even reading a book or newspaper was too painful. By all means, use your voice, contralateral arm or legs and feet to take over some repetetive tasks.

However, you need to use, strengthen and stretch your wrists as well. Targeted massage and strengthening with a physiotherapist should help. As workoholics we often forget how to listen to our body. Pain tells you to take a break, not do an allnighter. Then when having pain, it is easy to try to "protect" the affected bodypart, putting undue stress on other parts of the body, while the muscles, tendons, ligaments of the affected bodypart continue to weaken because they are unloaded and/or not used throughout their physiologic range of motion.

Before I jump to SSRIs I would start with psychotherapy, meditation, taking a real vacation, a healthy anti-inflammatory diet (standard food pyramid will do as a start), sleep hygiene to get 8h+ for recovery, sports (both endurance and strength, preferably something involving your arms and hands!), whole body sports massage etc.

I did find that the pain was to some extent in my head, to some extent real, and exacerbated by both overuse and chronic misuse.

"All in your head" describes the entirety of your experience in life.  All pain is experienced in the brain, as is pleasure, anxiety, etc.

Most experiences have a causal link from the outside, and the question isn't "whether" but "how much" should be externally mitigated vs "just" perception/attitude adjusted.  "just" in quotes because that's often the more difficult intervention.

RSI is pretty real-world, for many people at least.  No clue if yours is, but it seems reasonable to experiment on the easiest parts first.  Ergonomic setup of work space, exercises, attention to posture, etc. are the first things to try.  Simple rest probably isn't sufficient if you're not examining/understanding the causes of injury.

In parallel if you like, or afterward if you're apprehensive about it, talk to a psychiatrist about your anxiety or other symptoms - not just the wrist pain, but why you think it's psychiatric vs physical, and what other difficulties it's causing you in life.

First, pain in the wrists is often to due to muscle knots (“trigger points”) in the forearm muscles that you may not be aware are there until you go probing for them. There are many online resources for treating such knots if you find them. My advice: don’t overdo it, or you’ll bruise yourself and make it worse.

Second, the main thing you should not do if it is psychosomatic is take pain medication. Your brain and body can easily become psychologically and physiologically dependent on even “benign” drugs like NSAIDs, leading to situations where you’ll be in pain when you don’t take them.