It happened rarely, most notably a contingent of KV-1 tanks in Leningrad in August 1941 and perhaps later. (Source: Salisbury, Harrison The 900 Days: The Siege Of Leningrad) The factories in Leningrad never entirely shut down although some machinery was evacuated to the east.

However, this was a rare and desperate measure. In general, for nearly all armies during WWII, tanks were transported over long distance by rail or ship. Even the most mechanically reliable Tanks during WWII broke down on the march quite often. So driving them for long distances would mean a lot of tanks didn't arrive at the front in a timely manner or in a combat ready condition. Even a march of 100-200km would produce non-trivial tank losses in Soviet tank units, especially during 1941 when teething troubles on the T34 and KV tanks had not been worked out yet.

In addition to being literally true occasionally, the idea is very much true in spirit. Soviets had to improvise in 1941 and 1942 and their armored units were typically thrown into battle very quickly after formation. As the Soviet situation became less desperate in 1943, they typically were more careful about throwing armored units into battle too hastily. Giving troops time to train together and get used to their equipment, tactics, and supporting arms greatly increased their effectiveness and Soviet armored units accordingly became much better performers as the war went on.