To my mind three factors come into play in this question.
(1) The traditional rabbinic citation form is a product of the schools which post-date the fall of the Temple. I think this is the most important. Although "rabbi" is used (e.g.) in the gospels (Matt 23:7-8, John 1:38, etc.), it is not used in the same way as the rabbinical schools and authorities known from the Mishnah onwards.
I quote (briefly! there is more in context) from G. Stemberger (in a work revising that of H. Strack), Introduction to the Talmud and Midrash (T & T Clark / Fortress, 1991), p. 4:
From the modern perspective the year 70 is a [sic] unequivocal turning point in Jewish history. ... The introduction of the title 'Rabbi' (to be distinguished from 'Rabbi' as a form of address meaning 'my lord, my master') suggests such a consciousness of a new era. This is reflected in t.Eduy 3.4 (Z. 460): 'He who has students who in turn have students of their own is called Rabbi'. ...
This chain of authority (and authoritative citation) is not yet in place in Paul's day.
(2) On two occasions Paul is eager to show that he does have an authoritative train of tradition behind him. The first is in explaining proper order for the keeping of the Lord's Supper in 1 Corinthians 11:23:
For I received from (παρέλαβον ἀπὸ) the Lord what I also delivered (παρέδωκα) to you, that the Lord Jesus on the night when he was betrayed took bread,...
Here the formula of "reception and delivery" is what corresponds to the citation formula the question is interested in.1 The same pairing crops up a little later in this letter, 1 Corinthians 15:3:
For I delivered (παρέδωκα) to you as of first importance what I also received (παρέλαβον), that Christ died for our sins according to the Scriptures...
On reflection, it is interesting that both of these occurrences fall within this letter to the church at Corinth.2
(3) We note also in (2) that Paul's source is Jesus. He wasn't interested in other sources, and to cite an authority would run totally against this self-understanding. As Paul boldly states in Galatians 1:12:
For I did not receive it from any man, nor was I taught it, but I received it through a revelation of Jesus Christ.
Summary - For these three reasons then (chronology, comparator, and Christ, we might say) we don't find Paul using the later rabbinic citation formula.
- The lexical form of the words in this pair are παραλαμβάνω (paralambanō,“receive”) and παραδίδωμι (paradidōmi,“deliver”). I don't know whether it's significant that the only other place in the NT that this pair comes together is in John 19:16 -- of Jesus being handed over for crucifixion -- but I doubt it.
- Paul spent a long time in Corinth, but it was also a place where, it appears (most apparent in 2 Corinthians) that Paul's own authority was somewhat suspect there:
- 1 Corinthians chs. 1 and 3 brings the factions in the church aligning behind various leaders to the fore;
- in 1 Cor 2:1-3, Paul describes his own weak oratory -- not simply a "rhetorical" device, as it finds an echo in 2 Cor 10:1.
- And, of course, 2 Cor 10-12 has this (Paul's "authority" in Corinth versus his "rivals") as its central focus.
And it is only in his writing to the Corinthians that Paul uses the formula of "reception and delivery". Possibly not a coincidence, then.