Greek had been a very important language in the eastern mediterranean before the rize of the the Roman empire and continued to be widely used in the region even after centuries of Roman rule. With the collapse of the western half, the eastern Byzantine had Greek as their majority language.
While the germanic people that invaded and established their kingdoms in the western half adopted Latin (which later evolved into various romance languages) due to its prestige and how well established it already was with the population, the people who later settled in Byzantine lands didn't adopt Greek.
I can see why the Arabs and Ottomans had no desire to assimilate the Greek culture that they conquered since they were enemies to them but is there any reason as to why the Bulgarians, Serbs and Albanians didn't welcome it the same way the Germanic people embraced Latin in the west? Did they significantly outnumber the Greek speakers in the region even after centuries of Greek presence in the area?
While we can't say for sure as not many sources survive, demography seems to have been the key.
First of all, inscriptions of the late antiquity show that Greek had been spoken southwards of the line Durres-Skopje-Sofia-Veliko Tarnovo prior to the migration of the Slavs. While this line (called "The Jireček line") should be taken carefully, as the communities had mixed and many enclaves existed, the population that lived north to this line generally spoke Latin and not Greek.
Primary sources show that the Slavic immigration was not immigration of simple small warbands, but that the Slavs brought entire families with them. At the same time, the population from both rural and urban communities migrated towards Thessaloniki and Constantinople. But the Slavs didn't start this process: Ostrogoths, Huns, Langobards, Gepids and other have been ravaging northern Balkans as well. The Slavic immigration was far bigger than the Bulgarians, Serbs and Croats: Slavic communities migrated to the southernmost parts of the Peloponnese. They don't seem to have met meaningful resistance. There are many toponyms bearing Slavic names such as balta (march), ezeros (lake), goritsa (mountain peak), kamenikos (stony peak), lanka (ravine), zagora (behind the woods). See this map, which shows the number of Slavic settlement names per 1000 km^(2) . Local sources have been adopting Slavic loanwords, and there is even appearance of Slavic nasal vowels in local Greek terminology (Longos, Mesolongi).
However, the Taktika of Emperor Leo VI shows how his father (Emperor Basil I) started the process of reintegration by making the tribes leave their traditional customs and made them Greek (the verb γραικόω was utilized). The Slavs were integrated by baptizing, recruiting into army, and giving their chieftains Byzantine titles and honors. At the same time, many of them kept their internal autonomy: the Melingues, which lived between Taygetos and Oitylos in the Peloponnese, even though they were made Greeks, maintained their autonomy all the way until the Ottoman conquest.
There aren't - or at least I don't know of any - explicit mention of the language shift of these Slavs, but the very act that there were no Slavic speakers in most of the mentioned areas when the Ottomans arrived shows that they were linguistically assimilated.
However, the Slavs which organized into separate countries were an entirely different case. The Byzantine court has employed another very flexible strategy: giving the Slavs their written language. Allowing the Slavs to keep their language and customs proved to be far more fruitful strategy than to force them to accommodate to the Greek way of living. A useful counterexample can be found in Croatian litoral, where the Glagolitic priests insisted on using their local language well into the modern era, even though Rome insisted that they give up on this practice, sometimes resorting to even declaring them heretics.
