Does he, though? The way the periodic table is laid out is, in part, a deliberate attempt to impose beauty on the order of the elements anyway. There were plenty of other ways it could have been laid out. Some, arguably, more sensible than Mendeleev's choice, or at least more reflective of the underlying nature.
But anyway, that such beauty can emerge naturally from natural laws is extremely relevant and is reflective of the fact that the stunning complexity of some systems often belies the simplicity of how they can be described. This simplicity at the heart of nature should be viewed as an/ the answer to any argument from design.
And for that matter, beauty is in part what we choose it to be. I don't think there is anything that could be called inherently or universally "beautiful". There can always be opinions, or different perspectives, and different definitions on what counts as beautiful. The Periodic Table as it exists today is far more a sign of what we think of -- and in particular what Mendeleev thought of -- as beautiful than it is a sign of some Intelligent Creator.