Boyer-Lindquist is used for a rotating mass not for a non rotating mass.
Using Schwarzschild coordinates there is only one r coordinate. Eddington-Finkelstein introduces an additional r* called the tortoise coordinate. With this in place we get tilting cones.
And if one additional coordinate is not enough we can always use the Krusal coordinates that introduces two coordinates that relate to t and r. In Krusal coordinates the cones are always straight up and also 45 degrees (so no narrowing).
A very good question!
For starters it would not be applicable for Schwarzschild coordinates since this is a view of the situation from the perspective of a distant observer far removed from the gravitational field. So the point is that in Schwarzschild coordinates the observer is fixed.
But with regards to a observer in free fall who is approaching the event horizon and continuing towards the singularity, the speed of light will remain c, at least locally, so that implies that the angle of the cone would remain constant. However we can hardly speak of a cone
from the observer's perspective, it would seem clear that since the curvature for this observer is so strong that a construction of an orthonormal coordinate system would show anything but a cone for incoming and outgoing light rays except for a very small local region. Past the event horizon the observer would still measure the speed of light at c, again only locally, but because the curvature is getting so strong here that the closure between the observer and the center of mass is faster than the width angle of the observer's light cone would allow. Now some seem to interpret this as time becoming space and vice versa, which is the whole point of this topic.
To me that is not a coincidence at all. Remember that the Schwarzschild model is a view from the perspective of an observer far removed from the gravitational influence.