Warm rinse likely was a concession to those who still stuck with soap on wash day. Ivory Snow and some other products were still out there in late 1960's that were pure or mostly soap.
Warm water does give slightly better rinsing (detergent or soap)and extraction, which is why commercial laundries rarely rinse in cold water.
Much would depend upon what temperature "cold" water is coming out of taps. During warmer months, especially hot summers around here tap water is around 85 or bit above degrees F. OTOH once temps drop and remain chilly tap water is quite cold indeed.
SQ washers at local laundromat take hot water into FS dispenser and maybe one side, with cold as balance. Know this because one sees steam wafting up from dispenser as machines fill for final rinse.
In any event warm rinses virtually vanished from domestic washing machines as energy crisis of 1970's rolled out.
Finally warm water final rinse does lessen energy used by tumble driers or ironing (by hand or machine), and or pressing. Federal government, Consumer Reports and others pushing "energy saving" use of cold water for rinsing admitted as much, but would go on to say amount of energy saved was "minimal" compared to what was used to heat water for warm water rinsing. In their considered opinion cold water rinsing was just as good.