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The extra bar gives the band more time to adapt to the tempo, rhythm and feel. More bars are added as the tempo increases (or whenever the band leader deems necessary). So for instance:

1-2-3-4 (slow)
1-2 1-2-3-4 (medium/fast)
1-2 1-2 1-2-3-4 (very fast)

The extra bars are counted differently to avoid confusion and make it clear when to actually begin playing (always after the 4 unless the song has a pick-up). Imagine you would count-off 1-2-3-4 1-2-3-4. Your band members wouldn't know if you are using a single or a double beat count-off.

Why do dancers only count 5-6-7-8?

Because they don't need to get the tempo etc. from the counting, only when to start.

It's a mystery.

Could it be that the accompanist actually maintains the best?

My friend Google suggested this:
https://www.nytimes.com/2001/04/15/nyregion/in-person-never-stop-moving.html?searchResultPosition=1

My friend, the NYT, will not let me read the article without a subscription

Why wouldn't they need the tempo? They dance in time to the music, right?

Because they get thd tempo from the mudic, snd the beat is maintained by the music. Just a thought without evidence.

And, to answer the question why they don't count a single measure 1-2-3-4: dancers think in double beats since they often combine two beats for one full pattern. For instance, in LA style salsa dancing (which is the most danced salsa style worldwide), the basic pattern is:

1 - left foot forward
2 - transfer weight
3 - LF back to starting position
4 - break

5 - right foot backward
6 - transfer weight
7 - RF back to starting position
8 - break

(this is for leaders, followers obviously reverse the movements)

So one pattern is 8 beats long. That's why dancers count to eight. A lot of dance music reflects this by having phrases of two measures long.

That's correct.

@Montuno
Thanks much!

Montuno, you got ahead of me. On leading musical instruments, only fingers and hands move, the movements of which fit into the usual counting . The dancer has inside the loop doubled the number of moving limbs, which naturally influenced the counting.

I answered this.

Ah, yes, I'm familiar with this sort of choreographer. I expect it went:

One two three four five six se-ven and eight () ()

making up two bars of 6/8.

Interestingly, the version that goes on in my head is:
a 1, a 2, a 1, 2, 3, 4. Anyone else with me?

In my first piano lesson in the New year, my teacher, who is a performing musician. The first 1 2 is to set the half note/minin beat, and the last 4 count is to set the quarter note/crochet note bear. This makes more sense to me.

And what if it's in a time signature other than 4/4?

This is just an example of how to get everyone on ‘the beat’. Seriously, I don’t need to spell all the time signatures for you?

You don't. My point is that it's not really a helpful way to count-in and seldom do I see it used.