I'd like to start with a caution against two things.

The first is a general methodological one: counterfactuals (e.g., "why didn't something occur") are basically impossible to prove. This is further compounded by the fact that spoken languages don't leave material evidence, making confirming hypotheses about what did happen with the Indo-Europeans difficult enough.

The second is a more specific warning not to conflate the Yamnaya culture with Proto-Indo-Europeans, and especially not to conflate them with the Indo-European migrations in general. While the members of the Yamnaya culture did very likely speak an early Indo-European language and they did live in the area of the Urheimat at a time when we would expect Proto-Indo-European to be spoken there, whether the language the Yamnaya culture spoke was Proto-Indo-European and whether it was descendents of Yamnaya culture who proliferated during the Indo-European migrations are both still open for debate. Let's put a pin in that second part; I'll come back to it later.

To give a general summary of the speculation (and it is speculation) as to why the Indo-Europeans didn't proliferate into modern-day Georgia and Azerbaijan, let's put ourselves in the leather shoes of those Indo-Europeans on the Pontic Steppe, just beginning to spread. To the west is a vast swath of fertile grasslands and forests stretching all the way out to the Atlantic. To the east is a vast swath of fertile grasslands and forests stretching all the way out to the Pacific. To the north is a smaller swath of less fertile grasslands and forests stretching into the Arctic. And to the south is a very tall mountain range. Of all the directions to go, I would not fault the Indo-Europeans for not making south their first option.

Mountains are also not the best places geographically for people and languages to spread, with adverse conditions allowing groups to remain isolated from their neighbours and, in turn, maintain high linguistic diversity. These areas are now often called "residual zones", after Johanna Nichols' work in Linguistic Diversity in Space and Time (the Caucasus is one of her main examples), and are juxtaposed with "spread zones" like the Eurasian Steppes, which are characterised by high population mobility and low linguistic diversity.

Of course, it's also entirely possible that Indo-Europeans, maybe even members of the Yamnaya culture itself, did cross one of the various passes that link the lowlands on either side of the Greater Caucasus Mountains, simply to die out thousands of years ago with ne'er a trace. For a while, it was thought that the Maykop culture (ca. 3700 to 3100 BCE on the Russian side of the modern-day Russo-Georgian border) and the Kura–Araxes culture (ca. 4000 to 2000 BCE in modern-day Georgia, stretching into the Anatolian and Armenian Plateaus) may have been Indo-European, as they both exhibit the archetypical Indo-European tumulus-style burials and have material connections to other contemporary likely Indo-European groups. One hypothesis, proposed by Gamkrelidze and Ivanov in a series of works in the 1980s, even locates the Indo-European Urheimat in the Armenian Highlands, but even that hypothesis sees the migrations largely avoiding Transcaucasia.

So, why the gap? How did Indo-European languages end up on either side of the Caucasus, but not in present-day Georgia or Azerbaijan?

To put it simply, in order to get from the Pontic Steppes to the Near East, groups didn't travel as the crow flies through the Caucasus, but instead either travelled west, rounded the Black Sea, then came back east, like the Hittites; or travelled east, rounded the Caspian Sea, then came back west, like the Iranians. The four relevant groups to discuss here are the Armenians, the Iranians, the Hellenes, and the Slavs.

Starting off with the Armenians, we don't know when or how they ended up in the southern Caucasus. There isn't even a well-accepted hypothesis. They were definitely in the area by 200 CE, and the Armenian language is first attested in 405 CE. Both Martirosyan and Diakonoff, based on what they consider the presence of Urartian (a non-Indo-European language related to Hurrian) loans in Armenian or vice versa (their models are conflicting, but not mutually exclusive), conclude that Armenian speakers were present in the Anatolian and Armenian Plateaus during the Kingdom of Urartu in the 800s to 600 BCE; Diakonoff has, at times, gone so far as to say that the Urartian was solely a written language and that Pre-Armenian was both the common and elite spoken language of the kingdom. Robert Drews in The Coming of the Greeks makes reference to a consensus that they entered the region from the Balkans, likely crossing the Bosporus, around the same time as the 1200 BCE Near Eastern Bronze Age Collapse (though the idea that they were related to the Phrygians, another Indo-European culture in Anatolia, comes from a throwaway line in Herodotus; the linguistics doesn't bear that out); on the other hand, the appearance of "Armina" under Indo-Iranian rule (though the Akkadian translation of the Behistun inscription indicates some continuity with Urartu) and the prevalence (at least among the elite) of Indo-Iranian names, language, and religion has led others to conclude that they arrived from the east with the Indo-Iranians around 600 BCE. The other option is that Armenian speakers crossed the Caucasus by themselves, though when and from where that would be is unknown.

The Indo-Iranians we are more certain about. They emerged from the eastern end of the Late Proto-Indo-European continuum some time after 2500 BCE (generally connected to the Sintashta culture; start dates are controversial, but it transitioned to the Andronovo culture around 1800 BCE) and rounded the Caspian Sea to float around in Central Asia (as the aforementioned Andronovo culture) for the next thousand years or so. From there, the Iranians split into two broad groups, with the Eastern Iranian Scythian cultural group moving back into the Pontic Steppes and the Western Iranian Medes, Parthians, and Persians moving into the Iranian Plateau, both around 800 BCE. Of note here are the Ossetians, a Scythian group who migrated into the Caucasus in the 600s CE and still live in the Greater Caucasus (mostly in Georgia and on the Russo-Georgian border, in a region called Ossetia). Also of note are the various Western Iranian groups (particularly the Persians and the Kurds) who would, and continue to, be present in the Southern Caucasus from about 600 BCE onwards.

Hellenes would enter the region a number of times after the Alexandrian conquests; most notably for our purposes are the various Pontic Greek groups, originally based on the now-Turkish southern coast of the Black Sea (which they reached by crossing the Bosporus from the Balkan peninsula), who would sporadically settle in the Caucasus under the Pontic, Roman, Byzantine, Ottoman, and Russian Empires, some of whom, mostly in Georgia and Armenia, continue to speak Caucasus Greek.

At last come the Slavic speakers; let's take out that pin from earlier. While the Indo-European migrations are most associated with the Bronze and Iron Ages and prehistoric groups like the Yamnaya, it's important to note that the Indo-European migrations never really ended; indeed, the modern distribution of the Indo-European languages in central Eurasia owes far more to fairly recent centralised administrative empires than it does to prehistoric steppes peoples. Slavic speakers entered the Caucasus region with the Russian Empire in the 1800s, long after the Yamnaya had died out, and weren't even particularly present in what is now Southern Russia until the conquests of Ivan the Terrible in the latter half of the 16th century. For most of recorded history, the area was primarily populated by Iranian-speaking Scythian groups and various Turkic-speaking groups like the Kipchaks and Khazars. Slavic speakers, at the time of your map, were living in the orange area on the Baltic coast and the Central European Plain (the Pomerian and Przeworsk cultures, among others), and would spread east and south as the Rus' did the same over the course of the European Middle Ages. As part of the 19th century conquests, Russian became the lingua franca of Transcaucasia, a status it retained through the Soviet period and to today.

In the end, the Indo-European migrations were not a single event 4000 years ago that set off, spearheaded by the Yamnaya, with the end goal of assimilating the Eurasian continent (I think we associate steppes peoples too readily with Genghis Khan), but were instead built off individual, and often disparate, groups of people travelling rather sporadically over vast swaths of land over the course of millennia, in a process that is, by some metric, still ongoing.