Wirings without relays don?t strain the wirings, that?s a wrong concept. Yes, heavy current has to travel a longer distance before its delivered to the headlights, obviously voltage drop and loses occur. But your car came from the factory without the relay, right. So performance should not deteriorate over period of time so point of failure cannot be a wirings but the dipper switch which has metallic contacts which loosens over period of time or carbon accumulates over it.
Members have advised regarding installing the relays, yes I agree as it should be installed and most new cars now comes with preinstalled relays but your problem seems to be with vehicles alternator. It may have few burnt diodes, maybe faulty voltage regulator, blown EFI fuse etc. first get your alternator checked before moving ahead with relays installation.