As for Centralians (Swinton), they won the Rotherham Charity Cup in 67/68, a success which presumably prompted them into moving up to the S&H County Senior Lge for the following season.
68/69 CSL Div 2 champions, and RCC winners {#2}
69/70 Rotherham Charity Cup winners {hat-trick}
71/72 Rotherham Charity Cup winners {#4}
72/73 Rotherham Charity Cup winners {#5}
75/76 Rotherham Charity Cup winners {6th win from 6 finals!}
79/80 listed as 'Centralians (Rotherham)' in Green Un's CSL Div 1 table, they were relegated. {Sheffield Centralians were also members!}
80/81 now known as 'Rotherham Club' after 1980 summer merger with Scholes FC, promoted from CSL Div 2
81/82 CSL Div 1, Montagu Cup runners-up
82/83 CSL Div 1
83/84 the new Whitbread CSL Premier Div = 3rd/14, League Cup runners-up {to Swinton Ath} https://en.wikipedia.org/...ll_League#League_Cup
84/85 CSL Premier = 13th/14, relegated
85/86 CSL Div 1 = bottom/14, relegated
86/87 CSL Div 2 = 9th/13
87/88 CSL Div 2 = 8th/13
88/89 CSL Div 2 = bottom/14, folded ?

I think Centralians played at the LNER ground on Rowms Lane, but cannot confirm this. Perhaps they chose the name 'Rotherham Club' to try and emulate Sheffield FC who are/were universally known as 'Sheffield Club' ?? Whatever the reason it didn't work, after the fairly successful 83/84 season it was downhill all the way to extinction!