I'll assume you're a native English speaker. Since English doesn't have these clusters, it's difficult for an English speaker to hear or produce them correctly. But it is not impossible, and there is no reason to think Ancient Greek speakers did not pronounce them.
Word initial plosive-plosive clusters are definitely possible since they occur in modern languages
It is quite possible to have word-initial clusters of plosives without an intervening epenthetic vowel. Polish is an example of a language where this occurs; the word for bird is "ptak." Here is the Forvo page: https://forvo.com/word/ptak/
To me, it sounds like the female speaker is saying "tak," and the male speaker is saying "puh-tak." But this is probably not what they are actually doing.
I would guess that in fact, she is using a pronunciation with less audible release of the initial /p/, while he is giving it more audible release. It's true that it's hard to hear a stop with little audible release when there is no preceding vowel. But that doesn't necessarily mean the speaker is not articulating them. Aside from being easily to hear word-medially, as you mention, it would likely be more audible when it comes at the start of a word, and the preceding word ends in a vowel. This is a common situation in actual speech; speakers don't often say words in isolation.
While audible release sounds like a schwa in this position to an English speaker, phonetically there is a difference between audible release, and a vowel such as schwa. Hearing a schwa is an auditory illusion, like how Japanese speakers hear an /u/ vowel between the consonants of word-initial consonant clusters in English, such as the "s" and "m" of "smile." (In fact, there are good phonological bases for both of these illusions: Japanese speakers commonly realize /u/ as zero in some positions, such as between two voiceless consonants, and English speakers may realize schwa as zero in some positions, such as between two voiceless consonants as in "suppose" or even in other contexts as in "police". But these are language-specific processes.)
I don't know enough phonetics to be able to say how easy it is to distinguish aspiration word-initially before other obstruents. In fact, though, I don't think Ancient Greek has such a contrast; there might be exceptions that I can't bring to mind, but I think before unaspirated plosives we only see π τ κ, and before aspirated plosives we only see φ θ χ.
Demotic Greek shows reflexes of the first consonant in such clusters; this indicates that there was also a pronounced consonant in the earlier forms of Greek
LjL's answer makes a good point about these clusters being present in modern Greek and in loanwords from Greek in other languages. I left a comment where I wondered if these could be spelling pronunciations, but then I realized I could probably look that up myself to some degree. Wikipedia indicates that Demotic Greek, which is the form of Greek that evolved mostly on the spoken level (so it was probably not influenced by the spelling) does have reflexes of these sounds. For example, "φταίω" is the demotic pronunciation corresponding to the learned πταίω.
So in fact, we could say that even the Greeks somehow "preferred" other types of clusters over plosive-plosive clusters, and in unlearned pronunciation the first plosive naturally evolved to an aspirate or fricative, but the fact that this happened is evidence that the initial plosive was never lost altogether.
(This just applied to plosive + plosive clusters — unaspirated stops were unchanged in demotic Greek before nasals like /n/ or fricatives like /s/).
I think this, along with the fact that the Ancient Greeks must have had some reason for writing these clusters of consonant letters in the first place, makes it pretty clear that the first plosive of word-initial clusters was pronounced in Ancient Greek.
