Whenever a light source is directly seen in a reflection, it gets rendered as a bright white orb. This happens with both Point and Spot and Area lamps, sun lights seem to be the only ones not affected. Is there no way to get rid of this render breaking artifact? I'm assuming this must be a known bug with Eevee / Screen Space Reflection as it's been around for many Blender versions.
I solved the issue and am happy with the results, even if this is technically working around it. The core problem was that I was using a huge light radius: You want the sphere of the point light to encompass and represent only the bulb. I set it to 0.01 instead of the default 0.25: This not only results in the correct shadow softness, but the white orb is now just a tiny speck, precisely how a bright bulb should appear in a reflection! It's also hidden behind a lamp cover so it shines even this way only when seen from the right angle.