As he approached 80 my father, a large man, found that a medium-light .243 Win worked well given his declining strength/shoulder.
Noting that the .243 Win cartridge is simply the necked-down .308 Win thus having similar case capacity, yet substantially less sharp recoil plus excellent deer-size game ballistics. Weight, balance, and shot-type (brush/open/rest/no-rest/careful aim or jump/snap-shots) are probably large factors in your selection, as well.
https://www.wideopenspaces.com/8-gre...ints-and-more/
Alternatively, with/without recoil pad:
https://www.gunbroker.com/item/907274442
https://www.gunsamerica.com/90610134...VER-ACTION.htm
https://www.iammo.com/guns/rifles/ca...n/lever-action
Alternatively, if you are hunting in medium/thick brush, perhaps a Marlin 336 in 44 Rem Mag
https://www.gunsamerica.com/91636927...-44-Magnum.htm
https://www.gunbroker.com/item/907388118
https://www.gunbroker.com/item/907378505
https://www.gunbroker.com/item/907364037