That’s because the M113 is not in-game, at least players can’t operate it, M163 VADS is relatively heavier than the base M113 variant which requires more preparation for amphibious operation than BMP-1 like sealing the turret ring, something that BMP-1 doesn’t need and M113 doesn’t have a turret to be sealed. @Jadenbetter-live arguments falls short since all military vehicles need to go under safety check before operations,
BMP-1 isn’t special on that regard, such things like valve intakes, engine parts changing also are done before operations, not in the field. Regarding the turret position, the only moment that it needed to be positioned in the user manual I’ve found since it wasn’t detailed nor explicit said which manual, what pages and so on I couldn’t check myself, is the loading procedure of the cannon, which is required to elevate the gun in +3°:
To load, the autoloader arm hinges upwards, and the grenade is pitched forward over the gunner’s right shoulder and into the breech. To load, the cannon must be elevated by +3 degrees.
(Source)
Since anything is done before operations, return the M163 VADS amphibious capability but its turret must remain stuck for the whole time during the match if the player wanted to drive through a deep body of water, considering there are no field modifications in the game, yet.
See also:
The arguments used in the video as why the BMP-1 wouldn’t be so effective in water doesn’t correlate in War Thunder: water conditions, soil condition, crew experience, vehicle and parts maintenance and preparation are not in-game, yet if it’s ever considered, maybe never will be.

The YouTube video uses the Handbuch für mot.Schützen II Schützenpanzer. 1. Auflage as primary source, the image above attached are the listed equipments that is needed for the function of the vehicle in amphibious operations, most of which are already inside the vehicle and part of those that the crew can operate without leaving the vehicle.