Not generally understood that the Beeching report "Reshaping of British Railways" just looked at whether services were financially worthwhile. It made no judgement on whether they were worth having in a wider sense. That came in a subsequent report after all the hoopla, and led ultimately to the 1968 Transport Act which put subsidy for defined loss-making services on a statutory basis. Prior to this the government stipulation was that the railway had to break even.
Liverpool to Southport was a good example of a line with not only excessive peaking, but other issues as well. Like some other isolated DC lines it had a very expensive electricity tariff from the Central Electricity Generating Board. Running over low level sandy ground which had become outer suburbia it had a lot of level crossings manned 18x7. It had to shoulder the bulk of the costs (because it had the bulk of the passengers there) of the big, expensive termini at both Liverpool and Southport. The Ribble bus company ran cheap and extensive parallel services which prevented increasing the fares by much. Fares in those times were still on the statutory pence per mile basis but where this was uncompetitive with buses etc there were various workarounds, such as Cheap Day tickets valid all day for commuting.
Beeching was also a Wake-Up Call to BR management who were just letting things slide along as they were. Little attempt on cost control - apart from the absence of a fireman on the trains, manning was pretty much at Victorian era levels. As elsewhere, there was ludicrous over-provision of first class accommodation on the electric trains, which ran around almost all of the day as empty stock. These things needed sorting out.