Typically the Byzantine Empire was sorta politically 'there' at the time of 1356 through out 1453 - they did rises and falls in territory but really as a "realistic" thing they could not see the light of day unless the Ottomans experienced Ultra Civil War, level 10 difficulty.
But EUIII and EUIV allow the player to play any nation and see what he can do or she can do with it. People in Vicky II can try to make the god-damn Omans a civilized nation, or play as a Revitalized Greece who seeks to take over the Ottomans and resurrect the Byzantine Empire.
This should be the conclusion: In weeaaal life was there hope for Byzantium? Likely not, then again stranger things have happened (Hannibal invades the Roman Republic and just taunts the Romans to see if they could go out to face him. Really surprising that not only did the Romans beat him (Hannibal) but also won the Second/Third Punic War.).
But in game? Why not allow it? By this logic any nation that is either small or surrounded should not even hope to succeed. Meaning you can not play as the natives. In the end you die, whuchu gonna do bout it? You can't argue with the force of the Imperialist Powers backed with aggressive colonists seeking to take over America, in the end when the Colonial Era starts and begins to really kick in you have to say "I give up" and let Europe take over.
Or any of the French vassals. You can't break away or try to create France in your own image. I'm sorry, you just can't.
Nor can you form Russia as Ryzan or Nizidiy Novgorod. Don't even try anything as Smolensk!
I mean really this kinda defeats the purpose of Europa Universalis. "what if?"
By this logic in CK2 you should only win as William the Bastard. When you play as Harald or Harold II, you can't win. You can only delay the inevitable of William the Conqueror winning due to his use of calvalry, arches and infantrymen as a single force.
Recognizes what, PRC or ROC?
Well, what do you think? :V That whole thing in the 70s..I dunno whut is history again?
The US shifted its recognition from ROC to PRC for cold war political reasons,
When China began to oppose the USSR. You can say the reasons, they are rather simple and eloquent for the US. For topic: The US would support a Byzantine Emperor. Real truth. I mean they supported a tyrannical monarchy during and after WW2, so bad to where Harry S. Truman had to say "Well not all states are perfect..." as even the worst anti-communists were applauded by Greece's actions towards those who fought for Greece's independence from Nazi Germany.
In 1945 everybody recognized ROC.
That is mostly due to the ROC was suppose to be sorta a collective regime between parties. It didn't, and really the 'control' was rather vague due to the support of warlords and the ROC putting the Chinese Communists on their to do list rather than the Japanese invaders.
Had ROC remained in control of most of the mainland and the PRC was off in the boonies on Soviet life support, likely nobody outside the Soviets & Warsaw Pack would recognize PRC (maybe not them, IIRC in 1945 USSR still recognized ROC from before Mao's revolution really got off the ground).
Mao's Revolution got off the ground during the Long March, it went into full-gear during the late 40s and succeeded politically in 49. Also Warsaw Pack didn't exist in 45... >:V
And the USSR held a political line of the Communists have to work with the Nationalities. Stalin reportedly told Mao "I'm glad you proved me wrong."
Also Mao's CCP was a hell-of-a-lot more popular with the denizens then the ROC. Same with Byzantium and the Turks - the Greeks made mythos about god-damn Constantine XI. The 'Marble King' to free Greece from Turkish subjugation. And really the Greeks sought to recreate a pseudo-Byzantium with their 1920s War.