What is the strongest wood glue for crown & casing joints?

That's because, this situation is end-grain to side or end-grain by definition. End-grain glues poorly. Don't design your application to rely on end-grain being glued. Cope the crown for interior joints. When I can hide it, I'll cross tack an exterior corner.

No glue stains well. The trick is not to choose the right glue for staining, but to keep the adhesive off of the visible surfaces (don't overglue, prefinish or use tape to keep squeezeout of fthe show surfaces, scrape clean).

For paint, the above still applies but with more margin of error. Choose your adhesive based on the material to be glued and the application (does it beed to be water resistant?, gap filling? pot life?, cost?), not the surface finish.

-Steve