My experience is that you must have a specific benefits need to apply for a PSC. We tried to get one as an easy route to proving PPS (to buy prize bonds). Appointment made, all necessary documents produced, but my wife and I were both refused a PSC. The person issuing them said that as we didn't need one to apply for a benefit now, she couldn't issue either of us with a card. My wife will be eligible for free travel in just less than 2 years and was told to come back a month or two before age 66. We were actually quite surprised at the policy. I can't imagine many countries turning away volunteers for what is effectively a national ID document.
Paul