The point is that being closer to the star when that happens doesn't provide you with more forewarning than if you look at it from home.

I don't think anyone is advocating that we send actual probes to Betelgeuse or IK Pegasi. I'm confused why one would think that would even be on the table. Even if we sent a probe today at a tenth of the speed of light (well beyond our current capabilities) it will still take around 1500 years to get to IK Pegasi. I don't know why one would even think that would be at all in the useful category.

What is helpful is having more space based observation equipment in our solar system. The more we put into space the less of a problem we have with atmospheric interference, artificial radio sources, and general light pollution. To use one specific example that would help a lot, if we had a series of optical telescopes that were spread out around the solar system we could use parallax measurements to get a better idea how far away Betelgeuse is. For a variety of reasons there's a lot of uncertainty about how far away it is with 330 light years as a lower estimate and around 700 as an upper estimate although it seems like around 640 is where things seem to be settling down at. Given the inverse square law for radiation, this matters for a supernova concern. A difference of 300 light years corresponds to about a factor of 4 in the radiation strength. Overall, most of the interesting, practical investigation and reduction of astronomical existential risks can be done right here in our home system.