I think the trouble with this idea is that Burks is only in the booth right now because they need to cover for Remy's unplanned absence. Burks might be the choice solely because he's available and willing. So no real chance to train anyone. Really, there's no real training to be had. This is a job you can't really practice effectively. You just have to jump in and swim.
True - yet I think there must be no shortage of former major leaguers who'd like a check for being in the booth?
As for practice, I'm of the mind there's not much in human experience (especially dealing with communication) you can't practice. There's no reason ESPN can't pay a guy or two to spend a week going over past broadcasts and cues with Burks. Other color-men: examples of what to do, what not to do. His own stuff: where he nailed it, and his worst couple of things to work on/avoid.
1) Get him a list of talking points to think about, and spend some time actually doing them to a recording or two of a game. Probably about 95% of coaching-commentary can be repurposed as color commentary. How to stuff: approaching a pitcher. Difficulties of batting. Situational hitting. What players are or need to be looking at at any given moment. Generally, it's all the backwards looking "just happened" stuff: what's a good play, what's a better play or worse play than what you saw - why that might or might not be reasonable to expect.
2) Do another run with "put a reasonably intelligent 8th grader in their shoes stuff." The various roles on the club. How it feels to be a rookie.
That's really all you need as a fill-in, but you can go for:
3) Predictive/forward-looking commentary, likely stat-based, such as noting shifts and talking about expected outcomes. Or commenting on pitching sequencing while it's gong on and what you expect to see. That's more advanced and requires folding real time analysis into your comments. (Eck is great at this, BTW.)
4) Lastly there's the anecdotal. Everyone can do it, so it's really a strength to be fought. Less is more, etc. But I honestly think if you've got the rest of the framework working, your anecdotes will tend to shine more.
What you don't want is obvious stuff only, like, "Man, a hit to tie the game sure would be nice." Or does-not-add anecdotal, like, "I'll never forget the first time I stole a base in extra-innings."
Anyway, all that can be thought about and practiced.
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As a somewhat related thought, Eck's great not only because he's colloquially saucy, but because he's a generalizer.
He can:
1) look at something specific (a play, or a pitch, or a sequence),
2) admire it for itself,
3) make the coaching comments,
4) relate it to other stuff, and
5) come up with associations/anecdotes.
His expressions/anecdoes get remembered, but the way he uses them is perhaps more important.