Well I'm going to tackle some of the assumptions that you make in your question before getting to the meat of it all. There's a great deal in your question that I think you skip over a little too quickly and it is really important to get a grip on the background of this situation.

The first thing to understand is the succession following Edward the Confessor's death, all of the events of the Norwegian and Norman invasions are tied up in this.

Edward the Confessor left no direct heirs. You mention his distant relation Edgar the Ætheling, but he was not crowned king. We can spend all day talking about why this might have been, he was too young, the Godwins wielded far more influence because of their extensive holdings and wealth, and I've seen an argument that the English nobles wanted to ensure a stronger leader given the controversy around Edward's succession.

The English witena gemont, or assembly, thus proclaimed Harold King, and this was about as legitimate as it could get within England. Furthermore, Harold DID have royal connections. His mother was the sister in law to Cnut the Great, and his sister was Edward the Confessor's wife, so it was not as if he came out of left field as a claimant or that he was not deeply involved in the politics of England.

However, there were others with claims as well. William of Normandy maintained that he had been promised the throne by Edward the Confessor if he died without an heir. Now this claim is obviously somewhat suspect, but it could very well have had basis in fact. if nothing else, the Norman presence and influence in Edward's court was felt keenly by commentators at the time and shortly after. However, this could have been played up in the post Conquest era to further legitimize William's claim. Either way though, it cannot be dismissed out of hand as fabricated. The Norman biases and connections between Edward and Normandy are certainly present in the textual evidence available to us.

However, even ignoring the issue of whether a Norman bishop did promise William the throne on Edward's behalf, William and Edward were also related. William and Edward were first cousins through Edward's mother, Emma of Normandy. So he could claim some connection to the throne as well.

So even with just these two claimants to the throne I don't think there is a basis to decide that Svend had the strongest claim. Harold had both royal connections and the support of the English nobility. William had just as immediate claims as Svend did.

I'm not going to spend much time on Hardraade because I don't really know a whole lot about him, and I think overall he is pretty secondary to the content of your question.

Now before I get to the direct Danish involvement, I'm going to cover a bit of the situation in post-Conquest England. With the deaths of Harold and his immediate obvious successors in Hastings, the Anglo-Saxon nobility was thrown into chaos. There was an abortive attempt to crown Edgar king, but this wasn't able to stop William from marching on London. In the immediate aftermath of this, several Anglo-Saxon nobles rose up in rebellion and the campaign to put them down would become known as the Harrying of the North" which I have written about before.

I think it is worth bringing up some of what I wrote there because it is relevant to your question.

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William and Oderic were writing approximately a generation later from the events themselves, but they were quite clear in their condemnation of his actions at this time. Vitalis had this to say about the events themselves:

The King stopped at nothing to hunt his enemies. He cut down many people and destroyed homes and land. Nowhere else had he shown such cruelty. This made a real change. To his shame, William made no effort to control his fury, punishing the innocent with the guilty. He ordered that crops and herds, tools and food be burned to ashes. More than 100,000 people perished of starvation. I have often praised William in this book, but I can say nothing good about this brutal slaughter. God will punish him

Now it is impossible to fully quantify this and medieval estimates at casualties are nothing if not exaggerated, but the dire picture that he paints is matched by William of Malmesbury's description:

He then ordered both the towns and fields of the whole district to be laid waste; the fruits and grain to be destroyed by fire or by water, more especially on the coast, as well on account of his recent displeasure, as because a rumour had gone abroad, that Canute, king of Denmark, the son of Sweyn, was approaching with his forces......Thus the resources of a province, once flourishing, and the nurse of tyrants, were cut of by fire, slaughter, and devastation; the ground, for more than sixty miles, totally uncultivated and unproductive, remains bare to the present day. Should any stranger now see it, he laments over the once-magnificent cities ; the towers threatening heaven itself with their loftiness ; the fields abundant in pasturage, and watered with rivers : and, if any ancient inhabitant remains, he knows it no longer.

So the picture that the contemporary sources is quite bleak indeed. The question then is can we rely on these accounts or are they exaggerating the scale of the destruction? The other question is how much of this devastation is due to raiding by the Danes and Scots who were also active militarily in the region at the same time?

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link with sources

Now I brought up at the tail end of that question the possibility that Danish and Scottish raids and military action played a part in the devastation of the North.

The Anglo-Saxon Chronicle says that the Danes launched a raid on England and started dealings with the English before sacked a monastery that had been staffed with Frenchmen. After this they met with William and going home with more treasure, before being scattered by a storm.

We do not get any indication from this source the amount of money that William gave the Danes to go away. By this point in history, paying raiders and looters to go away had a long and storied history with varying degrees of efficacy. It was a strategy that had even been employed by men such as Alfred the Great.

The wording in the Chronicle is also ambiguous as to why the Danes showed up in the first place. It can be read as just a very large raid that the English went along with or that they too were trying to establish dominion over the island.

As for why the Danes didn't just take the money and keep fighting? Any answer here will be pure speculation so I'm not going to spend too much time on it. All I'll say is that conquest is risky, tribute in hand isn't.

In the same year came King Sweyne from Denmark into the Humber; and the landsmen came to meet him, and made a treaty with him; thinking that he would overrun the land.

link to the online translation I'm using

This general timeline of events is very similar to Orderic Vitalis's account in his history of England and the Church, but he does not mention William and the Danes coming to terms and leaving for tribute. His account comes across much more as the broader rebellion failing and the Danes cutting their losses and leaving after being defeated in several areas by William. However, he does say that the Danes were originally inspired to launch their invasion because of Svend's claim to the throne as well as at the behest, and pay, of the English. The Danes were just unsuccessful in their attack according to Orderic Vitalis.

In the interest of time and space I am not going to cram in the account that Vitalis provides in here, if you're curious for the exact working I'll put it in a response. Suffice it to say though, he does not leave a lot of room for ambiguity here. In his account the Danes are decisively defeated, even too scared to venture ashore for fear of the inhabitants.

This would thoroughly disprove, Wikipedia's assertion, shocking I know. Instead of having a favorable position, the Danes were starting to get scattered before the Normans and very few of them ever managed to get back to Denmark from the combination of losses to the Normans as well as to the elements. This would also indicate why they didn't just take the money and keep on fighting. They quite simply weren't in a position to be doing anything aggressive at the time.

Both sources agree that the Danes were there with the explicit support of the native English as a part of their rebellion. It is unclear exactly what the Danish motives were for this given the disparity between the sources. In general I am more inclined to lean towards Orderic Vitalis's account.

Now just to end, Svend's claim to the throne was really no weaker than any of the other claims. His mother was sister to Cnut the Great after all, and the Danes had extensive relations with the English going back centuries at this point.

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Sources:

I've used the Anglo-Saxon Chroncile and Orderic Vitalis's History of the Church for the majority of this answer as far as primary sources go.

for secondary sources and further reading,

Mason, Emma. House of Godwine: The History of Dynasty

Hale, Betty. "History of Prince Edgar & his Claim to the English Throne"

Donald Henson, "The English Elite in 1066: Gone but not Forgotten"