The Optimizer's maths is right — and beside the point
Let me concede the headline numbers up front, because pretending they're wrong would be silly.
A Class 3 voluntary contribution costs £18.40 a week, or £956.80 a year. It buys 1/35 of the full new State Pension — £241.30/35 = £6.89 a week, or £358.50 a year. Break-even in 2.67 years of receipt. Twenty years of receipt at the basic rate of income tax delivers around £5,736 net. The Optimizer is not making this up.
What the Optimizer is doing is pricing the contract as if it were guaranteed cash, when it's actually a contingent annuity with three live risks: longevity, political change, and opportunity cost. Strip those out and the trade looks unbeatable. Price them in honestly and the picture changes.
A SIPP isn't an annuity contract. It's an asset wrapper. You own the units. You decide the asset allocation. You can take it earlier, leave more to your family, draw it down faster in low-income years and slower in high-income years. The Class 3 contribution does none of that. It just pays you a small monthly amount, conditional on your continued existence and on the Pensions Act 2014 staying intact.