The traditional explanation is not correct. The phrase evaṃ me sutaṁ is a discourse marker whose purpose is to indicate that the speaker was not present at the events in question, but is passing on second-hand knowledge. In other words, it states that what follows is an oral transmission.
The defining context is a passage that appears in MN 123 and DN 5.
In MN 123 Moggallāna describes various deities to Ven. Abhiya Kaccāna, who responds:
The venerable Anuruddha does not say: ‘Thus have I heard’ or ‘It should be thus.’ Rather, the venerable Anuruddha says: ‘These gods are thus and those gods are such.’ It occurs to me, venerable sir, that the venerable Anuruddha certainly has previously associated with those deities and talked with them and held conversations with them.
Thus the fact that he does not say evaṃ me sutaṁ is a sign that he was actually present.
The same usage is found at DN 5, where the Buddha narrates to some brahmins the details of a sacrifice in a long-gone age. The brahmins conclude that, since he did not say evaṃ me sutaṁ, he must have been present at the sacrifice.
Incidentally, in some translations you may encounter the opening of suttas phrased like this: “Thus I heard at one time …”. But the fact that evaṃ me sutaṁ is an independent idiom used in this way shows that is an incorrect translation.
Similar usages are found elsewhere. In AN 3.57, for example, Vacchagotta says to the Buddha:
sutaṃ metaṃ, bho gotama, samaṇo gotamo evamāha
I have heard, Master Gotama, that the ascetic Gotama says this:
Again, clearly indicating that it is something that he has heard people say about the Buddha, not that he heard it from the Buddha himself.
Similarly, at MN 56 we find:
Sutaṃ metaṃ, bhante, isīnaṃ manopadosena taṃ daṇḍakīraññaṃ kāliṅgāraññaṃ majjhāraññaṃ mātaṅgāraññaṃ araññaṃ araññabhūtan
I heard that the Daṇḍaka, Kālinga, Mejjha, and Mātanga forests became forests by means of a mental act of hate on the part of the seers.
On the other hand, if someone wanted to indicate that they had actually heard something from the Buddha, they used a different discourse marker. For example at AN 3.80, Ānanda says:
sammukhā metaṃ, bhante, bhagavato sutaṃ sammukhā paṭiggahitaṃ
Sir, I have heard and learned this in the Buddha’s presence
This idiom is widely used, and as in this example, it is often associated with Ānanda. Thus if the texts were preserving his voice, claiming to be a direct witness of the teachings, this is what they say.
The fact that our text acknowledge their provenance as oral literature is a good thing. They were honest. They are not trying to pretend to be something that they weren’t. By clearly stating their status as oral texts passed down through tradition, they remind us of the Buddha’s words of caution regarding such transmission.
What we have in the Suttas is not a snapshot or recording of events, it is a narrative and a teaching passed down through a community. This is why it’s important for us to try to understand how such processes work, and how they affect the texts in their current form.