To be sure, the answer to this question would be mostly speculation because the historical record is woefully lacking in primary sources which relate to this topic. In addition, primary sources related to this topic do not explicitly say that the Mongols understood the historical significance of the cities you mentioned.

Something we all need to be aware of is the Mongols world view. The Mongols understood the world in their terms. They felt that everyone should be semi-nomads, living in gers, foraging, raising livestock, and riding horses. This is much of the reason that the forces of Genghis Khan destroyed irrigation, farmland, and buildings during their early conquest. It is very likely that the Mongols knew about the West, they certainly did during the later years of the various nation-states. As far as taking these cities for a strategic advantage is concerned, yes. The Mongols would have taken these cities if there was a strategic advantage. It is not likely that they would have taken them with prior knowledge regarding their importance.

What I can tell is that the Mongols did have connections to the West, which I'm sure you already knew. Several famous chroniclers traveled to Mongolia, including William of Rubrick, John Plano de Carpini, Juviani, and others.

We do know that the West had very limited knowledge of the Mongols before the thirteenth century. David Morgan argues that Genghis Khan's conquest of Central Asia and China went mostly unnoticed. He also argues that the first person to travel to Mongolia and return with accurate information was Friar Julian of Hungary in 1234. The invasion of Kievan Rus' lands, undertaken by Batu and the Golden Horde (this is the collective name), alerted Europe to the oncoming threat.

The Ilkhans under Hülegü, brother of Mongkë Khan, had interactions with the Papacy and the Crusaders. In 1260, the Ilkhans found themselves facing intense pressure from the Mamluks in Egypt, to the West, and their kin, the Golden Horde, to the North.

David Morgan states:

the Ilkahns of Persia....speedily dropped their old hauteur and began to see the Christian powers as potentially useful allies. The basic idea behind such projects for alliance was a combined operation against the Mamluks: a crusading force would be sent from Europe and its activities would be co-ordinated with an Ilkahnid invasion of Syria. Should Syria be conquered by the allies, the Crusaders would again take possession of Jerusalem.

Negotiations went back and forth between the two, but ultimately, little came from the agreement. The Ilkhans were pushed back by the Mamluks, and someone specializing in the Crusades will have to take up the European story from here. This is one example of a clash that involved one of the cities you mentioned. I would argue that explicit plans to take one of those cities as a gateway to the West were not in place during the reign of the Mongols. Strategic plans for taking those cities as a strategy to subjugate the West may have existed.

All of the places you've mentioned were historically important, but did the Mongols understand this? We simply don't know. We can certainly imply and suggest that they were not aware of the historical importance of these cities in religious or geopolitical terms.

I hope this is of some use to you and all those reading.

Suggested Readings:

David Morgan's The Mongols, 1986.

J. J. Saunders' The History of the Mongol Conquests, 1971.

Timothy May's The Mongol Art of War, 2007.

Document Reader - Morris Rossabi's The Mongols in Global History, 2011.

The mongols ( and turks/central asians/etc ) had contact with christians/muslims even before much of europe. There were mongolian tribes who converted to christianity ( nestorians ) since at least the 7th century. There were mongols who were christian, muslims, etc. Not only that, the mongol empire had innumerable christians, muslims, european, arab, jew, persian, etc. The mongols certainly knew of these cities.

The pope sent delegations to the mongols since at least the early 1200s. One of the first western histories of the mongol empire was by italian friar Giovanni da Pian del Carpine, whom the pope sent to genghis khan with a letter "Cum non solum".

The pope even advocated for a franko-mongol alliance against the islamic caliphate, but the mongols weren't interested in religious wars as they were a religiously inclusive empire and more focused on wealth and trade.

Neither Jerusalem nor Mecca would have been of any interest for the mongols in terms of conquest. The mongols, as I said, were after wealth, not religious symbols. Rome and constantinople would have been of interest. Any hopes for constantinople was pretty much destroyed when the turks in anatolia successfully broke off the mongol empire and created the ottoman empire ( who themselves would go on to conquer constantinople ). As for rome, the roman leadership ( the papacy and the business elite ) were more than willing to cooperate with/submit to the mongols and as a result was given preferential trade opportunities and favors within the mongol empire and as a result the italians grew extremely rich which in turn set the stage for the italian renaissance. The relationship between the italians and mongols were so fruitful that the italians started to name their children after the mongol rulers such as Hulagu Khan, who was a nestorian christian.

"The Story of the Mongols: Whom We Call the Tartars" : Fr. Giovanni Carpini (translated by Erik Hildinger )

"The Mongols and Global History" : Morris Rossabi

"Genghis Khan and the Making of the Modern World ": Jack Weatherford

While I don't have an answer to your question. I am interested in knowing why you've included Mecca in cities important to Europe when it was never a political capital or a centre of power