Cryptics from the perspective of an American satirist

Question

One segment of a certain satirical TV series began like this:

Good evening everyone. Today I'm going to rant about cryptic clues and cryptic crosswords – not our type, of course. Some say that Mike Pence is so interested in them that he neglects official business.

A message displayed beside him read, with the bullet point:

• Mike maker frantically finishing off order (4)

It's not just limited to America, folks! All around the world, wherever there is English there are cryptics and they are popular, at least among the erudite. And they are usually set by devils, literal or otherwise.

• Right within Tom's love city (6)

In Britain, where such monstrosities were first conceived, during World War II you had to solve them fast in order to break codes at Bletchley Park. We don't need that to test the mental faculties of our people. And we have computers now!

• MI6 chief turned, greeting Ojibwe national summit (8)

While there is some software to automatically solve our kind of crossword – Dr. Fill, anyone? – as far as I know there is none for the cryptic type. But crosswords have no immediate application other than to "train your brain", whatever that means.

• Software publisher dismantling bonded burr (10)

Wait, why are the messages all cryptic clues? Production! Will you stop poking fun at how this segment's name is stylised? I can see the answers alongside them!

• Hide and speak for a punk band (12)

Gosh, I'm outta here. Cut the tapes, I'm facing a Formidable Opponent next.

The answers to the five cryptic clues above, as well as the segment's name, are all related. What is that relation and what is the TV series's title?

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_{Some of the clues may be unfair. If that is the case, I apologise, for this is the first time I've made them.}

Answer 1

Solutions:

Mike maker frantically finishing off order (4)
RØDE (anagram of ORDER minus its finish)
Right within Tom's love city (6)
TROMSØ (R inside TOMS O)
MI6 chief turned, greeting Ojibwe national summit (8)
MØLLEHØJ (M + HELLO< + OJ)
Software publisher dismantling bonded burr (10)
BRØDERBUND (anagram of BONDED BURR)
Hide and speak for a punk band (12)
LeATHERMØUTH (hide->leather, speak->mouth)

The common feature is

Ø

and I'm guessing that maybe this is

the segment called "The Word" but often written "THE WØRD" from the Colbert Report (which, as well as having the Ø, was fond of its bullet points), though that hasn't been on the air for some years now.