And amphibious crossing is not simple as just driving your vehicle into the water.
The depth and width of the river, bank conditions, and the current’s velocity are major factors to determine the unit’s ability to conduct a crossing, even if amphibious. These factors will determine if the force can cross by fording/swimming, and by rule, reconnaissance of the near shore and predetermined crossing sites are necessary. Such actions are performed, whenever possible, by engineer reconnaissance teams which might be drawn from the divisional level, identifying "wet gaps" early and deploying the required resources to allow hasty crossings.
Also, due to the limited organic crossing assets, additional support from echelons above the brigade combat team or regiment is often necessary. That support is only available when that HQ have taken purposeful action to position the assets at the right time and place and integrate them with the manoeuvre force to make a hasty crossing feasible. Coordination of these assets must be made early in the planning process.
So, as Veitikka mentioned, it is a complex process, both doctrinally and in game terms. Currently, we do have shallow water and fording areas which (at least abstractly) assume above-mentioned preparations were made. I don't exclude that in future we might expand river crossing operations.