So before starting with the responses I would like to state a joke which I had read a few days back.
Someone asked a lawyer that if he is watching Ramayana n Mahabharata on television. To this he replied that yes and I understood that Ramayana was an abduction case while Mahabharata was a property dispute.
Yes Mahabharata indeed was a property dispute of the highest order.
The direct blood lineage was never any contention because neither Pandu nor Dhritrashtra were biologically kuruvanshi.
Again it was no where necessary that the eldest born of the king becomes the next king because Devrat decided otherwise and no one objected his decision especially since by then none of the sons of Satyavati were born and no one knew if ever one would be born.
So now coming to the generation of Pandu and Dhritrashtra. Both of them were ready to become the king and were happy at this idea. None of them had an objection to become the king. So Ideally had everything been normal Dhritrashtra would have been made the king, however he was blind so taking into account his physical incapability the next in line for the throne, Pandu was handed over the throne, but it nowhere meant that the Dhritrashtra progeny had lost their claim on the throne. (Remember in Treta Yug Ramji had made Bharatji as his Yuvraj so it's no where necessary that only elder son of the king gets the throne that was a common practice but not a rule)
Bheeshm had actually felt bad for Dhritrashtra for he was being punished for a sin not actually done by him, so he wanted that his son succeeds the throne, that was a reason he got Dhritrashtra married to Gandhari who had the boon to beget 100 sons. He had felt that his son would be born early and things will get sorted. Had Duryodhan been the eldest Kuruvanshi of his generation, Bheeshm would have probably supported his claim. But that never happened
Anyhow meanwhile the Pandu killing episode happened and he decided to go for Vanwaas with his wives to repent. The throne needed someone they got Dhritrashtra sit on it, but he never had a proper Rajyabhishek. He was more a representative of the king that being the king himself, however he was more than Bharatji was because unlike the latter he was allowed to sit on the throne. He had all the rights which any king had( remember this point) only excluding his claim to the throne over Pandu, it was clear that if Pandu returned he would be made the king. Aside by now they were aware that Pandu can not have a son(probably they didn't consider the option of Niyog back then), so things seemed finally sorted Dhritrashtra would be on the king till Pandu returns. On his return Pandu will become the king and maybe after that Dhritrashtra's sons will succeed.
But then Gandhari got a long pregnancy and didn't deliver on time, instead during that time Pandu n family decided to go for Niyog and Yudhishtir was born before Duryodhan (in fact even Bheem was a day elder to Duryodhan)
Had Pandu returned there wouldn't have been any confusion, he would have become the king, but unfortunately he died, and never returned to claim his crown. Now Kunti returned to Hastinapur along with the Pandavas. The ones to return were Pandavas and not Pandu so Dhritrashtra was not needed to vacate the throne for them(it's a stupid argument people made and Dhritrashtra continued because Pandavas were kids, Yudhishtir was over ten years of age back then and people did ascend the throne in young ages. Three generations later Janamejay was made the king at the age of 5) he continued to reign.
Now coming to the dispute
1) This is clear that Pandu was the king and Dhritrashtra was just an acting king in his absence. But in his absence he had all the rights excluding that over the throne in case of Pandu returned. Nominating and selecting the next king or the Yuvraj is the duty and right of the king, Pandu never appointed Yudhishtir for it, so shouldn't this duty and right now be bestowed upon Dhritrashtra since he was given practically all the powers and rights of the actual king? Does deciding on the next Yuvraj come under those rights and duties? No one had thought over it and hence didn't decide over this earlier.
2) Yudhishtir was supposed to have a more rights because he was elder. But how to calculate who was elder? Yudhishtir was undoubtedly born earlier, but Duryodhan was conceived earlier (I don't buy this long pregnancy point but since we are discussing epic so will have to go with whatever is mentioned) so which among these constitute for being elder
3) If being elder made Yudhishtir more entitled for the throne, why not being the son of elder brother make Duryodhan one. Dhritrashtra was not given throne because his wasn't abled but there wasn't any such disqualification for Duryodhan. If we are making the point that Dhritrashtra sat on the throne only because Pandu left, we can not ignore that Pandu was made the king only because Dhritrashtra was blind, so again who had more rights on the throne?
As far as the capability is concerned as someone already mentioned both were considered equally competent till the Varnavrat episode. Infact the citizens of Hastinapur liked Duryodhan more since he was with them since childhood.
At that point of time division of the state was the easiest task although it was an escapist method but the best way out. However then the Dyut happened. Yudhishtir staked everything and lost. @Janki no it's not that Duryodhan had staked kingdom or family. In fact he couldn't have done that, kingdom belonged to his father and his father had rights on the brothers. Yudhishtir had right over his brothers since their father was dead else the right should have been with the father. The game doesn't go that way. To give ab example Suppose two people are playing A and B, both of them stake Rs.100, A wins the game, now A simply stakes the amount on the board (200), but B has to stake new 200, if A again wins now B will have to stake 400 and so on. Similarly since Yudhishtir was losing, it was always he who had to stake something to equate with the things already on the board like once he lost his chariot, that was now considered staked by Duryodhan so Yudhishtir had to stake his n his brothers weapons and so forth. Duryodhan didn't stake his wife and couldn't have staked his kingdom and brothers so this son is only on Yudhishtir.
Anyhow now coming to the second dice game, the rules stated the the one who is defeated had to remain out for 13 years-12 Saal Vanwaas n 13wa Saal Agyatwaas, however they hadn't decided which calendar to use while calculation of this timeframe. Arjun disclosed himself the moment 13 years passed as per his calendar but as per Duryodhan's calendar there were still a few days remaining (I think it was a confusion of 365 days year and 365.23 days year), while Bheeshm was right in saying that since the calendar wasn't decided hence Pandavas are right to be out after the passage of time by the calendar they used, Duryodhan said no his calendar must have been considered so a again a technical glitch. Legality is all about being technically correct and they had left many technical loopholes throughout
So as someone has already pointed out the claim wasn't unilateral like most believe, both of them were right in their claims and were equally capable. I guess even Supreme court would have taken years to come to a judgement for this
If we discuss how could have Duryodhan ruled after having done so many crimes, then others might say how could Yudhishtir turn out to be a good ruler after having let his family become slaves, when he did not have the power to chose between two wrongs and two rights or to keep it cool during pressure situation
Sorry for the long post but it needed one