As to your additional sentence, the comma before “telling” is not necessary. Let’s see why. {[Many people think] = independent clause (that ATC consists of) = noun clause/object of the verb “think” (a row of controllers sitting in front of their radar screens at the nation's airports) = noun phrase/object of the preposition “of” and complement (telling arriving and departing traffic what to do) = DEFINING PARTICIPLE PHRASE/it modifies the noun phrase "a row of..." Because it is defining, which means it is a reduced adjective clause, no comma is needed here as you may know by now. Here is your sentence: {[Many people think] (that ATC consists of) (a row of controllers sitting in front of their radar screens at the nation's airports) (telling arriving and departing traffic what to do).} Here is what you understood: “{[Many people think] (that ATC consists of) (a row of controllers) (WHO sit in front of their radar screens at the nation's airports) (NO COMMA) and ((WHO) tell arriving and departing traffic what to do).}” One independent clause here. The “who” is optional here because “and” WITHOUT the comma before it means that there are two “predicates”. The first predicate is “sit”, and the second one is “tell”. Both verbs are related to the subject “controllers”. If you want to put a comma before “and”, you need to write the sentence this way: “Many people think that ATC consists of a row of controllers who sit in front of their radar screens at the nation's airports, and CONTROLLERS tell arriving and departing traffic what to do.” WITH the comma before “and”, the subject “controllers” needs to be added. The sentence would be like so: “{[Many people think] (that ATC consists of) (a row of controllers) (who sit in front of their radar screens at the nation's airports), and [CONTROLLERS tell] (arriving and departing traffic what to do).}” Two independent clauses here. I hope this is clearer.

I’m afraid that there is no adverbial clause in this sentence. Let’s see that together. {}= entire sentences []= independent clauses ()= dependent clauses (noun/adjective/adverb clauses + phrases) {[Some researchers say] = first independent clause ((that) the results constitute compelling evidence) = noun clause (that telepathy is genuine).} = adjective clause (defining which evidence) {[Other parapsychologists believe] = independent clause ((that) the field is on the brink of collapse) = noun clause (, having tried to produce definitive scientific proof and failed).} = MISPLACED PARTICIPLE PHRASE. The participle phrase is a reduced adjective clause when it is placed AFTER the noun/subject WITHOUT commas, and it acts as an adjective. When it is placed BEFORE it, it is just a participle phrase WITH commas, and it acts as an adjective modifying a subject or a noun. The participle phrase can either be placed at the beginning, in the middle or at the end of a sentence. Here are four easy examples: Beginning: Eating their food, (COMMA) the cats are happy. It means that WHILE THE CATS ARE eating their food, they are happy. It is a participle phrase here. In the middle (non-defining): The cats, (COMMA) eating their food, (COMMA) are happy. It means that cats IN GENERAL, are happy when they eat their food. It is a participle phrase here, as well. In the middle (defining): The cats eating their food are happy. (NO COMMA) It means that some specific cats THAT ARE eating their food (now) are happy. It is a reduced adjective clause acting as a participle phrase here. End: I saw the happy cats eating their food. (NO COMMA) It means that I saw the happy cats THAT WERE eating their food. It is a reduced adjective clause acting as a participle phrase here, as well.

For your sentence, I could either say: “Having tried to produce definitive scientific proof and failed, (COMMA) other parapsychologists believe (that) the field is on the brink of collapse.”. Beginning of the sentence. Participle phrase. Or Other parapsychologists(,) having tried to produce definitive scientific proof and failed(,) believe (that) the field is on the brink of collapse.”. Middle of the sentence. Reduced defining adjective clause acting as a participle phrase if used WITHOUT commas, or non-defining participle phrase if used WITH commas. The commas are optional here; it really depends on the general context; it could either be defining or non-defining. If it is defining, it means that the “other parapsychologists” have tried to produce.... If it is non-defining, it means that “other parapsychologists” IN GENERAL have tried to produce.... In this context, I don’t know if YOU know thanks to a context that you haven’t posted here who the “other parapsychologists” are, so it’s for you to decide whether it is defining or not. Note that we can't put the participle phrase at the end of the sentence in your sentence; this is the reason it is a misplaced participle phrase. Here are your sentences: {[Some researchers say] (the results constitute compelling evidence) (that telepathy is genuine).} {[Other parapsychologists(,) (having tried to produce definitive scientific proof and failed)(,) believe] (the field is on the brink of collapse).} Or {(Having tried to produce definitive scientific proof and failed,) [other parapsychologists believe] (the field is on the brink of collapse).} Anytime! Let me clarify the situation. The subjects are not the same; “some researchers” and “OTHER parapsychologists” are different. The tense is the same in both sentences: “(some researchers SAY...CONSTITUTE...IS)” = present tense; (other parapsychologists BELIEVE...IS) = present tense.

I move my comment to Additional details sections.