The answer to this question depends on whether you believe you should allow or prevent inflammation during recovery. This article explains both sides of the issue.
Your trainer probably believes that some level of inflammation, extra blood flow, etc. is beneficial to recovery, and that a jacket can significantly affect that.
Inflammation begins after you do a heavy set of lifts. Your muscles are stressed during the workout, and this triggers inflammation as part of the recovery process. Regardless of how you feel at this stage, your lifting performance will be reduced.
Some people believe that this inflammation is necessary for proper recovery and adaptation. However, that review article argues that the science is not settled. Should you use ibuprofen to help yourself deal with inflammation and get back to the gym a bit sooner? Or does that hurt your strength gains by interfering with the natural recovery process? The author's conclusion is:
if anti-inflammatory interventions delay healing or blunt gains, they
don’t do it by very damn much. Conversely, if they aid return to function or actually promote gains, again, it’s not by very damn
It seems to me that if the effects of doing something as drastic as using anti-inflammatory drugs or ice are so inconsistent or insignificant to your training, then the effects of wearing or not wearing a jacket will be much less important.