Later I heard from a doctor that the recommendation was followed by a visible increase of rheumatism, including small children, so they realized it was not that good idea.

Could be a case of post hoc ergo propter hoc. Rheuma has many variants, but generally they are all either autoimmune or degenerative. Juvenile arthritis (JA), an umbrella term used to describe the many autoimmune and inflammatory conditions that can develop in children ages 16 and younger, doesn't include cold as a causal factor.

You can kill cartilage using cold, however I doubt the very cold showers reached the -20°C and below from the study linked. Even then, cold induced cartilage necrosis doesn't necessarily cause degenerative joint disease. It may however cause inflammation? Somewhat dubious, I'd be suprised if so.

Ah, the experiments they did back in the seventies ... unencumbered by ethics boards. Simpler times.