Many thanks again for your advice on repairing this cheese board. I have finally got round to doing it. In fact when I had cleaned off the old glue I found that the ends were very rounded and blunt, with hardly any interlocking fibres. I suppose it wasn’t surprising having been broken and glued twice. I didn’t think that a new join would be strong enough to cope with routing out the old spline. So I chiselled out the spline before joining, and created a dovetail shaped spline and socket (see photos). I then glued up with epoxy as you suggested, using a form made of plywood scraps to secure it while clamping.
I’m quite pleased with the result. It’s far from perfect, but perfect was never an option. I was especially happy with the figuring in the wood, which emerged strongly when I used a scraper to remove 5O years of cheese grease!