This is simply wrong. We know tugs towed barges between the Gauging Lock and the Thames, apart from anything else because we have photos of them doing so:
Thames-scale barges above the Gauging Lock with river tug locking up to the pool.
Tug pulling first of barge train from Gauging Lock, 2nd barge still in Lock
The two barges below the Gauging Lock, getting towed out to the Thames. This was 1982.
In earlier times Thames sailing barges would navigate the same stretch between the Thames and the pool above the Gauging Lock, even if they couldn’t proceed further.
Thames sailing barges alongside footbridge in mid to late 19thC.