There are actually the remains of a temple that was used to worship both Menelaus and Helen. The site is known today as the Menelaion, the temple of Menelaus, but it was also known as Therapne to the Ancient Greeks. Despite the modern name, it is possible that the temple was primarily for the worship of Helen, with Menelaus being her divine partner. For example, Herodotus calls the temple "the sacred precinct of Helen" (6.61). The earliest text that I know of that refers to the site as the Menelaion is Polybius (5.18), while Isocrates records that both Helen and Menelaus were worshipped there (Orations 10.63). The earliest surviving finds are dedicated to Helen, and it is not until the fifth century that we find surviving votives dedicated to Menelaus.

The site is beside the remains of a Mycenaean palace complex, which appears to have been considered the home of the Homeric kings of Sparta, with Pindar calling it the home of Tyndareus (Isthm. 1.32-5) and the Dioscuri were buried there (Nem. 10.56). It should come as no surprise that, with Mycenaean archaeological remains and the association of the area with so many Greek heroes, a religious site was established here. Such activity was happening all over Greece, with old Mycenaean tombs serving as places of hero or ancestor worship. You can actually see drawings of some of the finds and the layout of the temple here. The start of worship at this site (at least, the evidence of votive offerings) comes from the eighth century, roughly contemporaneous with the establishment of hero cults throughout Greece.

There is no evidence that non-Heracledae heroes were ignored. There is no reason for them to be either. I imagine you may have thought such heroes may have been shunned as they weren't Dorian (please correct me if I am wrong), but it is important to remember that the Heracledae, such as the kings of Sparta, were considered Achaeans, not Dorians (Hdt. 5.72.3; cf. Tyrtaeus fr. 2 West). Indeed, the Spartans actually went out of their way to retrieve and move what was considered the bones of these heroes back to Sparta, as with the bones of Orestes in the mid-sixth century (Hdt. 1.66-8). Pindar actually calls Orestes Laconian and says that Agamemnon was killed near Amyclae (Pyth. 11.16, 32), and Pausanias says that he was shown the supposed tomb of Agamemnon there (3.19.6) and that Orestes' son Tisamenus was also moved to Laconia (7.1.8).

The traditional explanation of this activity is that Sparta was pursuing a 'philachaean' policy, one designed to earn the goodwill of the non-Dorian states of the Peloponnese, possibly including some of the perioikoi, marking a shift from military expansion from the eighth and seventh century. Rather, Hall has suggested that this political move, while somewhat connected to ethnic concerns, was more an attempt to justify Sparta's growing hegemony over the Peloponnese (2007, 336 in The Cambridge Companion of Greek Mythology).