Key rate duration for callable/putable bonds

Perhaps I’m overthinking this, but I cannot wrap ,y head around this 1 apsect of reading #44 on Bonds with Embedded Options…

Why are callable bonds with a low coupon rate unlikely to be called…and why are putable bonds with a high coupon rate unlikely to be put?

This relates to LOS k in Reading 44. Your help is appreciated.

If you issued bonds paying a 1% coupon, why would you call them? So that you can issue new bonds paying a 6% coupon?

If you own a bond paying a 12% coupon, why would you put it? So that you can buy a bond paying a 6% coupon?

And what does this have to do with key rate durations?

Schweser states “Callable bonds with low coupon rate are unlikely to be called; hence their maturity matched rate is their most critical rate (i.e. the highest key rate duration corresponds to the bond’s maturity)”

Maybe it would make more sense to me if it didn’t just say “low coupon rate”, but instead said if the coupon rate is lower than the YTM or lower than the spot rate, then they are unlikely to be called? A company would want to call a 1% coupon if the current spot was 0.5%, right?

Quite possibly, although the YTM would be more important than a single spot rate. It would also depend on the call price, and the time to maturity.

Would the YTM be important for the issuer? I thought its cost of financing is determined by the market interest rate at the time of issuance and the coupon rate.

Well, I guess the YTM would be useful to see how the market is pricing comparable debt and hence deciding if keeping the current debt at balance or calling it…

So i understand why low coupon is unlikely to be called, but then why "hence their maturity matched rate is their most critical rate (i.e. the highest key rate duration corresponds to the bond’s maturity)” So confused, anyone?