I've seen some very nice kitchens without backsplashes, but my impulse is that they have advantages- even if you are willing to repaint the wall, which is admittedly a minor inconvenience, I believe another issue is containing water and debris. Reading stories of tear outs, and being able to write some of my own, you know its "what lies beneath" that makes a huge impact on the sanitary conditions of your food prep area.
Water can exploit any crack or crevice, leading to mold and rotting particle board, wall board etc. Food debris which falls between the cabinet and the wall ends up near baseboards will attract mice, rats and bugs- all of which are unsanitary and many of which are destructive.
At minimum I would ensure a very tight fit between cabinet and wall (not easy in an old house, but you have to love shims) and caulk the remaining space between the cabinet and the wall, just like the kitchen sink and bath areas.
Wow! I didn't know I felt so strongly about backsplashes! Thow some water on me! Just as long as it doesn't get between the cabinets and the wall...have I mentioned what that can do???