Later I heard from a doctor
Apparently, the people giving health advice at internet yet have to learn
... that anecdotal evidence is not reliable?
I googled "cold shower rheumatism" and found a study suggesting that exposure to cold water (in this case, winter swimming) actually relieves the pain associated with existing rheumatism. A plausible explanation of your doctor's observation is that cold showers made people who already had rheumatism more likely to report it to doctors.
I would certainly be wary of making children do it, but I actually have noticed that my tolerance to cold weather has gotten slightly better after cold showers (and it's noticeably lower on the days where I don't take a cold shower). The effect may be mostly psychological but it's still an effect.
found a study
A very weak study in this context, for multiple reasons: Somewhat obscure journal (impact factor of 1.06), end point was the swimmers' mood, which may also influence pain reporting, intervention was winter swimming, which is much dfifferent from cold showers in many ways (shower != sports activity), the pool of participants was structurally non-overlapping (children versus people who go winter swimming!), the pain relief is confounded by also feeling "more energetic, active and brisk", compared to controls who did not do that spor...
Thus spake Eliezer:
It seems that many here might have outlandish ideas for ways of improving our lives. For instance, a recent post advocated installing really bright lights as a way to boost alertness and productivity. We should not adopt such hacks into our dogma until we're pretty sure they work; however, one way of knowing whether a crazy idea works is to try implementing it, and you may have more ideas than you're planning to implement.
So: please post all such lifehack ideas! Even if you haven't tried them, even if they seem unlikely to work. Post them separately, unless some other way would be more appropriate. If you've tried some idea and it hasn't worked, it would be useful to post that too.