I'm not sure this is the complete answer but here is a point not mentioned yet.

Even if rake is "correct" such that trail is the same between one frame and another, the shallower head angle will induce more camber change for the same amount of steering input. In other words, the wheel leans into the turn. This lowers the wheel center.

With a "normal" bike, the non-zero trail means that the contact patch rises when the steering is turned, which really means that the wheel center falls, which means that gravity wants to pull the wheel sideways (i.,e. flop over). But now imagine two bikes, one with a vertical head angle and no rake, and one with a "normal" head angle but enough rake to give it zero trail. The flop-over contribution from trail is zero in both cases, but the camber induced by the non-zero head angle still means the wheel center will fall and thus "want" to flop over.

One final question is where the "extra weight at the front" is located. If it's in a handlebar bag it is in front of the steering axis. If you're talking about rider weight, it is obviously further back. So you'd need to be more specific.