I think "While the empire was still strong" is key in understanding the answer to this question. As much as I love games like CK, and they do generally do a very good job at the historical setting and weighting things so that they tend to follow historical trajectories, they are also simulations that don't quite capture the events of real history in their entirety.

If you consider the course of Byzantine history, and chart it from roughly AD 300 to AD 1450, in that period of over 1000 years there are very few instances where the Empire was not losing ground, and the few occasions of relative strength were spent trying to recapture lost territories. If you think about the game again, as a player you are often trying to make choices that are in your interests without necessarily being swayed by any particular sentimentalism. It can happen though. EU4 has the Steam achievement to reform the Roman Empire and lots of players attempt to do just that. The Empire was not too dissimilar in its ambitions. In the sixth century the emperor Justinian spent a lot of the empire's resources attempting to reconquer the Western Roman Empire. They did pretty well, recapturing Africa, Italy, and parts of Spain. Could those resources have been better used? Yes, very much so. The reconquest was costly in terms of men and money and in terms of recuperating either one of those things it was largely a failiure, but then that was not really the point.

Another angle on this that develops the point is how we see value. You mention that the lands are fertile and the nations there are not as powerful. There are reasons for this, and these are partly because the value of the lands are different too. The Byzantines would not have been able to do much with a lot of Steppe land. They were more interested in holding cities and being able to tax those populations. The land in the region you mention was largely inhabited by nomadic tribes with little in the way of urbanisation, and therefore the region does not hold the same pull as reconquering Italy or Egypt, for example.

Below I have listed a brief overview of each century and the main territorial goings on. You can see that there are very few occasions where the Byzantines could embark on offensive wars and when they do they tend to be to recapture former territories. Having said that, they do have some influence in the region you mention, so if the circumstances had been right, maybe they would have done just as you suggested.

The 400s. There is a little corner of territory, south of the Caucasus Mountains called the Kingdom of Lazica that is somewhat part of the Empire and for a lot of the time they have influence in the Crimea. For the most part though, during this period they are dealing with the Gothic tribes and the Huns. The Western Empire collapses. Not much time to go out on offensive wars.

The 500s. If ever there was a time to head north this would have been it but Justinian (527-565) would rather make an attempt to reconquer the Western Roman Empire. It is a fairly success attempt but the gains are not long-lasting. Land is gained in 591 in Iberia and Armenia from the Persians but the troops are better spent guarding the Danube from the Avars.

The 600s. A very busy time for the Empire and one that doesn't go too well for them. Teh Avars invade in Europe and the Persians in Asia. Constantinople is sieged. Heraclius manages to defeat the Persians but no sooner is this done then the Arabs invade and many territories are now lost. From this point on the Arab/Muslim empires are dominant in the region.

The 700s. Similar to the 600s. In the West the Bulgars and Avars are dominant in Eastern Europe and in the East various Islamic Caliphates are raiding Anatolia.

The 800s. Lots more defensive wars against the Bulgars in the West and the Muslims in the West. Some recoveries are made.

The 900s. More offensive wars to subjugate the Bulgars in the West. In the East the Islamic Empries are going through some flux and so there are some gains but mostly these regions gain a larger degree of autonomy.

The 1000s. There is a bit of a high point around 1025 but even this is mostly the recapturing of former territory. There are struggles in souther Italy, some degree of peace in south-east Europe, and then wars with the Turks. In 1071 there is a significant defeat to the Turks who start to settle in what becomes Turkey.

The 1100s. Again, some recovery at the start under Alexios I Komnenos (1081-1118) but mostly they are trying to hold on to what they have got. Which they just about manage. There is a lot of pressure from the Turks but there are also Crusaders to handle. Mostly they are heading to the Holy Land but they often pass through Byzantine lands.

The 1200s. This is a real nadir. Following the sack of Constantinople in the Fourth Crusade (1202-1204) a lot of territory is lost to the "Latins" and the remaining Empire is split into three smaller despotates (and a few other regions), they are mostly trying to survive but the Nicaean group do regain Constantinople in 1261.

The 1300s. At this stage the remaining empire is on borrowed time really. In the West there is pressure from the Serbians and Bulgarians. In the East from the Turks. By the end of the 1300s there is very little left except a coastal empire around the Aegaen.

The 1400s. The Byzantine Empire is pretty much a city state at this point. They are facing the rise of the Ottomans, who prove to be their eventual downfall, but they also never have the resources to do anything meaningful from a military perspective.

Unfortunately I am not aware of much specific scholarship that deals with this question but some general sources that might serve a good starting point would include "The New Penguin Atlas of Medieval History" (C McEvedy: 1992), lots of maps and gives an overview of the migrations during the period. "Romano-Byzantine Armies 4th-9th Centuries" (D Nicolle: 1992) is not particularly good but it does cover the military period under consideration, one of retrenchment and defense. "Byzantium" (J Herrin: 2008) is a good, popular overview from a Byzantine historian of the empire. I can submit others if required.