Confusion surrounding etymology of 'Burdock'

Solution 1:

Is it simply a coincidence that 'Arctium,' meaning 'Bear,' the animal, sounds like 'Bur' in 'Burdock?' (viz., NOT bear-dock, but bur-dock).

Yes. Sound similarity is not a useful tool to figure out etymological connections between words. Bear and burdock are not cognates in English.

For instance, all of these words differ from bear in one vowel sound, but are otherwise unrelated etymologically to bear:

beer bore bur bar birr byre ber

Even the verb bear, meaning carry, is unrelated etymologically to the animal bear. The OED eventually traces the verb bear to:

the same Indo-European base as Sanskrit bhṛ-, ancient Greek ϕέρειν, classical Latin ferre, Early Irish beirid (3rd singular indicative), Armenian berem (1st singular indicative), Albanian bie (imperative singular bjer), all in the senses ‘to carry, to bring’, and also (with secondary senses) Welsh †beru to flow, Old Church Slavonic bĭrati to gather, Lithuanian berti to scatter.

So sound is insufficient to connect the etymology of two words. So is meaning.

Even if you could find a small bit of similar meaning between the words, that isn't sufficient for showing that the two words are cognates. For instance. island and isle look similar in form and mean similar things, but have completely different etymologies: island comes from Old English ig-land, and isle comes from the Latin insula. Word pairs like this are called false cognates, and they're textbook examples of how two similar words can form coincidentally without sharing a common root.

So if the dictionary is telling you that the two words are not related, you're going to need better evidence than sound or meaning similarity to show that they're related.