"So, is there a way to change the view angle without affecting the aliasing?"

To understand how that is connected, just dump your shadow maps and take a look at them while changing that cutoff angle.

Since the shadow map camera field of view is derived from the spotlight cutoff angle, it's created to fill the square shadow map exactly with the circle thrown by that spot light for optimal usage of the available texels.
If you blur that you will obviously lose data on the edges where there is almost no data except for the border, which is explicitly kept to throw shadows because all data outside the spot light cone doesn't receive light.

If you increase the camera field of view angle while keeping the shadow map resolution constant the shadow map precision decreases and if your blur is pixel based, you blur a lot bigger region when increasing the field of view, which gives you the softer shadow as result.

The way to define a spot light cone to contain the whole scene is pure linear algebra. All you need is Pythagoras and a tangens when the spot light is outside the world's bounding sphere.

Note that a the cutoff range of a spot light is the half-angle of the cone, it should be in the range of [0.0, 90.0) degrees. A bigger angle doesn't create a decent camera for rasterization because it would try to map stuff behind the camera in front.