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 sports activity, the list goes on.

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.

Yea.