Not if the coupon-paying security is held in a non-taxable account such as an IRA or 401K, which is the logical place to hold coupon-paying securities.

Depends where you live, among other things.

But I disagree with you on one point: it wouldn't make sense to own a high-coupon security in a tax sheltered account*.

Securities with biggish coupons are generally lower quality long term investments, which are purchased only because of the large coupon.
You give up a good long run total return because you're getting regular cash coupons which (for whatever reason) you like or need, and figure it's a reasonable trade-off.
Consequently you'll only want a large coupon if you're going to spend it. Most tax sheltered accounts are for longer term capital appreciation, so it's not a good fit.

As mentioned, it's generally not sensible to choose a high-coupon security for long term capital appreciation.
Reinvesting coupons is a bad idea all around, and low- and no-coupon securities are almost invariably the better investments in terms of total return over time.

In case anybody wonders about my rule of thumb that high coupon securities are usually poor picks over time, a random spot check:
Of the companies currently covered by Value Line with at least ten years of total return history:
Average 10 year annual rate of total return of companies currently with no dividend: 9.30%/year
Average 10 year annual rate of total return of companies currently with high dividend: 6.18%/year (top half of dividend payers)
(In this particular stretch the low-dividend firms were the best choice, but that isn't necessarily always true)

At the extreme, common stocks with indicated dividend yields over around 8-9% have average negative total returns over time.
Those yields never last: either the stock price rebounds or the dividend gets cut. Usually the latter.

Jim

* An exception might be a post-retirement account where you're required to make regular cash withdrawals.
Or not, if you still want decent capital appreciation from it because it's big enough and/or your life expectancy is long enough that the rate of return makes a meaningful difference.