Everybody has a personal allowance. Roughly speaking, you don't pay income tax unless your income (including pension) is over that threshold. As the basic threshold personal allowance is circa £12K and the average state pension circa £9K you don't pay any tax. Even then you only pay tax on the portion that is above your personal allowance.
The people worried about being taxed more are those in the wealthiest bands. So ignore Sunak's alarm messaging, as you should be OK. No more than a couple of bob a month if push comes to shove for the the lowest 60% of the population income band.
Yup, the full state pension is currently just over £11k a year and the single person allowance is just over £12.5k.
As you, and others, correctly pointed out, the state pension is taxable but if it is your sole source of income then you wouldn't currently pay tax but then again you would likely be also in receipt of other benefits which may or may not be taxable.
Around 20% of men and 33% of women in receipt of the state pension have no private pension but the majority of pensioners do have to pay income tax.