Who propounded "All doesn't exist" in the Buddha's time? SN 12.15 says,


  'Everything exists': That is one extreme. 'Everything doesn't exist': That is a second extreme. Avoiding these two extremes, the Tathagata teaches the Dhamma via the middle


We know that by the Buddha's time, the early Upanishads propounded some sort of "everything is existence" or "everything is the Self" doctrine.

But who propounded the latter doctrine ("Everything doesn't exist") in the Buddha's time? Certainly not Carvakas or Lokayatas, who were simply materialists. So who exactly was propounding this during the Buddha's time?
Perhaps it's a bit too frank, I think the OP asked an invalid question. A doctrine of everything doesn't exist cannot be uttered. It's like, a dead man cannot say &quot;I'm dead&quot;, or you tell someone in the face &quot;I'm not here&quot;. Making up such question is just an indication the questioner trapped by the trick of the mind - we called intellectual capacity.
This is the same trick for making up the phrase like the &quot;groundless ground&quot;, the &quot;egg-less egg&quot;, or the &quot;valueless value&quot;... the list can go on and on. You are simply making up another seemingly wisdom-infused phrase &quot;non-existing existence&quot;. The Buddha has vivid and fun examples to dismiss, said, these people are talking about and debating on the &quot;horns of a rabbit&quot; and &quot;hair of the tortoise&quot;. These things are mere constructs, the mind's game (戲論).
Let me make it more prominent: how can a question be made if all doesn't exist? What is to be discussed about since everything doesn't exist?
In ancient India, at Buddha's time, the study of Hetu-vidya and Catuṣkoṭi are prerequisites of establishing any doctrine. With all doesn't exist the whole proposition simply cannot be established, no learnt master would propound it.

Note:
I have great doubt on the OP's quoted Pali Sutta SN 12.15, the accuracy of the translated wording &quot;everything exists&quot; + &quot;everything doesn't exist&quot;. Perhaps due to this misleading wording the OP asked the unnecessary question.
The corresponding Agama read as this (excerpt):

...尊者陀迦旃延...：「...云何正見？...」
佛告陀迦旃延：「世間有二種依，若有、若無，為取所觸；取所觸故，或依有、或依無。若無此取者，心境繫著使不取、不住、不計我苦生而生，苦滅而滅，於彼不疑、不惑，不由於他而自知，是名正見，是名如來所施設正見。所以者何？世間集如實正知見，若世間無者不有，世間滅如實正知見，若世間有者無有，是名離於二邊說於中道，所謂此有故彼有，此起故彼起，謂緣無明行...滅。」
佛說此經已，尊者陀迦旃延聞佛所說，不起諸漏，心得解脫，成阿羅漢。
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...Ven. Kātyāyana, &quot;...what is right-view?...&quot;
The Buddha told Kātyāyana, &quot;There are two fixations in
the world, if exist or if non-exist, due to attachment on contact. For
attaching on contact, either fixate on exist, or fixate on non-exist.
If one doesn't have any attachment, tames the mind made it not attach,
not dwell, not involve the self. Suffering arisen as it arises,
suffering ceased as it ceases. In it don't doubt, don't faze. Not
depending on other but known it yourself, that is
called right-view. This is called the right-view taught by the
Tathagata. Why is that so? The aggregating of the world is known and
seen as it is; if the world non-existed it is not existing now.
The cessation of the world is known and seen as it is, if the world
really existed it is not ceased existing now. This is called
avoiding the two sides teaching the middle way. It is what called,
this exists therefore that exists, this arises therefore that arises,
called saṃskāra (activities) caused by ignorance... cessation.&quot;
... ...
... ..
... .
(the quoted italic text is corresponding to the OP quote.)

The Agama is much clear, concise and precise. Reading SN 12.15 ties a big knot in the head got lost in the bushes.