Primogeniture was a feature in English Royal inheritance. However, succession to the throne was generally determined by the old King who would nominate a successor himself. (Source: Reassessing Anglo-Saxon England By Eric John)

In the case of Edward the Confessor that you bring up; Edward had no children and failed to make it clear his preferred successor, leading to the Norman conquest of England as put forward by Richard Baxter, who wrote, '[Edward's] handling of the succession issue was dangerously indecisive, and contributed to one of the greatest catastrophes to which the English have ever succumbed'

There is evidence that he gave his support to all the claimants, but the validity of these sources is questionable given their intentions are to generally legitimize each claimant's right to the throne.

Edward Aethling actually died in 1057, the same year he returned to England after being exiled in Hungary for most of his life. His son, Edgar, was also given the title Aethling which generally a term given to those of noble birth or who were in line for the throne, which gives some credence to the notion that Edward the Confessor wanted Edgar to be King. Although he did little to affirm this, if this was the case. Some sources also claim Edgar was very briefly proclaimed King, but never crowned, after Harold's death.

Harold Godwinson's claim was based on a promise made to him by Edward, on his deathbed, that the Kingdom would be put under his 'protection' upon his passing. Although whether this happened or not has never been proven definitively. This scene in the Bayoux Tapestry apparently shows Edward promising the throne to Harold, although it is impossible to prove if that is what is actually going on. It is possible that he was proclaimed King in the absence of a more suitable candidate and the looming threat of the Scandinavians and the Normans.

William the Conqueror claimed that Harold was sent by Edward to reaffirm that William would be the next King of England. William also claims that Harold swore on holy relics to support his claim to the throne (this was how William got the Pope's support for the conquest of England). This version is also depicited in the Bayeux Tapestry, although again, it is hard to decipher what it is actually showing. William is backed by two contemporary writers, William of Poitiers and Williams of Jumieges, though these are two well known Norman apologists, who may well have been instructed to create a version of events that legitimised William of Normandy. The whole situation of how Harold ended up in William's court and the oath is a source of much conjecture between historians but it forms the basis of William's claim.

The situation regarding Edward's succession is actually very interesting, with lots of twists and unknown quantities, if you'd like to read more (and probably better) on the situation I'd suggest these;

Van Kempfen, F.J, (2009), ‘The Mercian Connection, Harold Godwinson’s Ambitions, Diplomacy and Channel Crossing, 1056-1066’

Oleson, T.J, (1957), ‘Edward the Confessor’s promise of the throne to William of Normandy’

Garnett, G. (1986), ‘Coronation and Propaganda: Some implications of the Norman claim to the throne of England in 1066’

This is the first time I've answered something on here so I hope this was okay!