I believe the siding was added in about 1935 and was indeed added mainly for the Metropolitan’s use. There were quite a few Rayners Lane starters in the peaks many of which ran to Aldgate. These were traditionally formed of 1904-13 saloon stock but later were used with O/P stock. There were even some Q38 stock trains on the Met in the early years of the war which operated through to Barking.
I can remember getting Picc's from Rayners Lane in the late 50's, with my parents. We travelled from Uxbridge and always took the Met to Rayners Lane and changed for the Picc there, as the Uxbridge platform indicators always advised, out of peak periods.
I remember this particularly because, occasionally, the Met was an F stock and I loved those large, eye like, cab windows
"Decline in passenger usage started in the mid/late 1950s, so that from Monday 2 March 1959 the Monday to Friday midday off peak Piccadilly line service to Uxbridge was withdrawn with all trains reversing at Rayners Lane." Underground News August 1982 p.161 link. After progressive service decline, from 5 October 1974 Piccadilly trains only reached Uxbridge Monday to Friday peak periods.