Yes, the cross member, original or removable, is there to absorb the lateral forces and prevent the rear axle from deforming.
When stationary, there are no lateral forces and it therefore only absorbs those produced by the inclined suspension springs if the rear wheels support the weight of the car, the chassis should therefore not or very little. deform.
In any case, to put it in place you have to remove, at least partially, the lower arm supports and for that the wheels must be in the air. But to tighten the bolts definitively it is preferable to rest the voture on its wheels so that the chassis is in the position as close as possible to the one it has in operation.
Once replaced, you will have to adjust the geometry of the rear axle and if the cross member pulls or pushes, it varies the camber, hence the advantage of having an adjustable cross member.