Whole sign houses can "solve" the problem of a nebulous birth time if the ascendant is in the middle degrees of the sign and there's no way the birth time can be off by more than a matter of minutes. In that case, shifting the ascendant a few degrees won't change its sign, so no matter what, the first house is the same sign, and so are all the rest of the houses.
If the ascendant is very early or very late in its sign, it could be in another sign if the birth time is even a few minutes off, so in that case, whole sign houses could give you the wrong sign for the first house, and for all the rest of the houses, too. If the possible time frame for the birth is a range of hours, even a mid degree ascendant could have its sign changed with the corrected birth time.
But no matter what house system is used, the ascendant is an important point, where it is by degree matters, and that will be incorrect if an incorrect birth time is used. Maybe it will be off by a little, maybe by a lot. In whole sign, the ascendant isn't the first house cusp (unless it's right at 0 degrees 0 minutes of its sign), but it still determines the first house.