I like the bevel vertex operation and can see many uses for it going forward although, as pointed out by @FedPete, this particular case leaves you with two faces with 5 sides (which he solves below although applying a “to sphere” or a shrink wrap modifier (which have not been covered in enough detail yet!!!) will further help the solution as some points will be incorrectly co-plannar!!)).
While I believe most of blenders operations work well with quads, and all work well with triangles;
n-gons (polygons more than 4 edges) can cause many problems in modelling, rendering, modifiers and animation.
They should be avoided, eliminated or at worst hidden where they will have least impact on a model. Many thanks @FedPete. I should have known better!