Disclaimer: I have no formal training in biblical languages.
I raised a very similar question not too long ago on B-Greek. There are actually several questions related to these words. About the 4th post in my question I ask pretty much what you ask but the takeaway for me was that while there are different words for "end/ending" it is only by context that you can establish the referent and meaning. I refer you to the discussion at B-Greek to get the benefit of the various perspectives.
Given the context we can take it either way:
- when will the ending start?
- when will the age have ended?
I think that their expressed curiosity about the temple's destruction indicates that they would like to know:
- when the fireworks would start ("when will these things happen")
- and when they would be over ("and when will be the end of the age")
So I would say that the modern reading is too specifically trying to place "end" at the "beginning of the end".
Another legitimate option is to read it as an example of Hebrew parallelism where the second phrase repeats the first.
Unlike the B-Greeks I consulted who didn't see much difference in the actual usage of τέλεια and its cognates, including συντέλεια it seems BDAG says that it refers to the end point, not the end process:
συντέλεια, ας, ἡ (orig. ‘community contribution’ [the basic semantic component relates to someth. that complements someth. already present]; in various senses Pla., Demosth. et al.; ins, pap, LXX; En 106:18; TestSol; TestJob 4:6; Test12Patr; ApcEsdr 2:31 p. 26, 25 Tdf.; Aristobul., in Eus., PE 8, 10, 9[=Denis p. 219 ln. 11f; Holladay Fgm. 2, 55 p. 138]; Jos., Ant. 15, 389; 20, 262; Tat.; Ath., R. 12 p. 61, 31 ‘increment’; Theoph. Ant. 2, 31 [p. 176, 14] ‘posterity’) in our lit. only in a temporal sense: a point of time marking completion of a duration, completion, close, end (Polyb. 1, 3, 3; 1, 4, 3 al.; SIG 695, 13 [II B.C.]; POxy 1270, 42 [II A.D.] σ. τοῦ ἔτους; LXX; Iren. 1, 6, 1 [Harv. I 53, 2]) συντέλεια (τοῦ) αἰῶνος the end of the (present; αἰών 2a) age (TestBenj 11:3; TestJob 4:6) Mt 13:39f, 49; 24:3; 28:20. τοῦ αἰῶνος τούτου of this age 13:40 v.l., 49 v.l. τῶν αἰώνων of the ages (TestLevi 10:2; Did., Gen. 115, 19. Cp. in gnostic speculation σ. καὶ ἐκπύρωσις τοῦ παντός Hippol., Ref. 9, 30, 8) Hb 9:26. τῶν καιρῶν (Da 9:27) Hv 3, 8, 9. τοῦ κόσμου Mt 13:49 D. ἐπʼ ἐσχάτων τῶν ἡμερῶν τῆς συντελείας in the last days of the consummation (of the age) Hs 9, 12, 3 (cp. TestZeb 9:9 καιρὸς p 975 συντελείας; ApcEsdr 2:31 τὴν ἡμέραν τῆς συντελείας; cp. ViJer 13).—DELG s.v. τέλος. M-M. TW.
Arndt, W., Danker, F. W., Bauer, W., & Gingrich, F. W. (2000). A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed.). Chicago: University of Chicago Press.
Exported from Logos Bible Software, 1:16 PM March 11, 2019.
I'm not qualified to say so I won't opine.