Is churning an effective way of saving money?

I think you are misinterpreting what churning is. As you have stated it is to "take advantage of promotional offers" (emphasis mine). That is, offers that go above and beyond the normal card rewards of a few percent cash back, or similar marginal rewards.

I just checked and American Express is offering $150 or $200 (depending where you look) cash back with their Blue Cash Everyday card if you spend $1000 in the first 3 months. That is 15% or 20% cash back on your first $1000 spent. If you are trying to gain benefit by credit card churning you would sign up for this card, spend $1000, or just a tiny bit more, and never use the card ever again. Eventually the company will close the card, or you can cancel it after any signup bonus imposed time limits have lapsed. You cycle through a bunch of these offers each year and can make a few thousand dollars, assuming you haven't spent money you wouldn't otherwise have spent. It does come with a slight hit to your credit score, if that is something you care about.

The purpose of "credit card churning" is not for the single digit percentage rebates on purchases.

Many credit cards offer a sizable bonus if you spend "X" dollars in "Y" months. The best offer that I have seen (and done) is $200 if you spend $500 in 3 months. Many lists of offers are available with a google search along with bank bonuses as well. For example, Chase Bank just brought back a $600 bonus if you deposit $15k in a money market account and $25 in a checking account, both for 3 months. There are some additional details but that's the gist of it.

I have 3 insurance bills per year which I can charge, each exceeding $500. If bonus cards are available, I open 3 per year and take whatever bonus they offer. Add a couple of bank rebates and it's easily $1,000 to $1,500 per year. I keep the cards until I find a new card and then close one for every one that I open. There's a small temporary hit to my 800+ credit score but I couldn't care less about that since I have no need for credit nor will I ever.

I help to support a disabled cousin and her blind husband so if the credit card companies and banks want to give me free money and make a small dent in that endeavor, three cheers for them! Apart from the 5 minutes it takes to fill out the application, the only other effort needed is to carry a different piece of plastic in my wallet.

As far as the cash back is concerned - The cash is a fixed percent and therefore will be different depending on your normal monthly spending. 20 years ago, I got a 2% cash back card that deposited to a 529 college savings account. With no other deposits, and including the 20 years of market returns, the account has just over $42K today. It will pay for the last 3 semesters of her college. The card has no annual fee.

If there are no cards that match up to your spending or lifestyle, it's ok to ignore this and move on. As Bob noted, there are bills I get such as insurance, that add up fast, and running the expense through a card for cash back makes sense. Even my charitable donations are more easily paid via card (and many charities will tell you they have a high cost to handle a check and prefer a card payment) and getting another $300 back in addition to the tax saving is worth it.

(The title question references churning, but you asked about general cash-back as well, and that's what I addressed.)