Comma separation dilemma

Your first instinct is correct. The conceptual rule you're applying is most easily described this way: The comma before the "and" is an indicator that what follows the "and" is referring to something earlier in the sentence. That is, "other fungi in this family" is referring to "the fungus", not "hardwoods". Following this rule will help you know when commas should appear in large sentences, and when they should not.

That comma should stay.

Your second instinct, to add additional commas around "traditionally", is an attempt to correct the flow of the sentence when read, by inserting additional pauses so that "traditionally" is given separate emphasis. It is correct insofar as it doesn't break any rules, but it's also a warning sign: Sentences that force the reader to pause too often are irritating to read and can disrupt comprehension. If you feel the need to insert commas just to slow down the reader, or add emphasis where they naturally wouldn't, then you are better off reforming the sentence.

So take your second instinct seriously, but consider a range of solutions, rather than just adding the commas.