Well, FWIW, here is a complete listing of early Jewish and Christian passages that use the expression:
Judges 18:10 LXX, 19:19-20 LXX, Nehemiah 6:9 LXX, Psalm 33:10 LXX, Ecclesiastes 1:15 LXX, Luke 21:4, 1 Thessalonians 3:10, 1 Corinthians 16:17, 2 Corinthians 8:14, 9:12, 11:9, Philippians 2:30, Colossians 1:24, Testament of Benjamin 11:5, 1 Clement 38:2, Shepherd of Hermas, Vision 3.2.2, Parable 9.27.2, Acts of Thomas 149.
I looked up all of these texts and the metaphorical sense of "lack, want, poverty, shortcoming" is what is found in each case, not the literal "coming late, a lagging behind". Each discusses either need in contrast to wealth or completeness, or shortcomings in contrast to moral purity. This sense is especially foregrounded when husteréma/husterémata/etc. is the object of pléroó or anapléroó "fill up, supply", as it is in most of the non-Septuagintal cases (e.g. 1 Thessalonians 3:10, 1 Corinthians 16:17, 2 Corinthians 8:14, 9:12, 11:9, Philippians 2:30, Colossians 1:24, Testament of Benjamin 11:5, 1 Clement 38:5). In these, anapléroó more naturally relates to the metaphorical sense of husteréma (supplying what is lacking), as these examples show:
"And when I was with you and needed something, I was not a burden to anyone, for the brothers who came from Macedonia supplied what I lacked" (2 Corinthians 11:9).
"He shall supply what is lacking of your tribe" (Testament of Benjamin 11:5).
"Let the poor give thanks to God because he has given them someone through whom his want may be supplied" (1 Clement 38:5).
I haven't checked the verb hustereó but the Liddell & Scott entry suggests a similar pattern; the literal sense of "to be lagging behind, come late, come too late" is found in the non-Koine classics (e.g. Herodotus, Thucydides, Xenophon, etc.) whereas the metaphorical sense "to be in want, poverty, have shortcomings" is found in Jewish literature (cf. Josephus, Antiquities 15.6.7) and in the NT (cf. Mark 10:21, Luke 15:14, John 2:3, Romans 3:23, 1 Corinthians 8:8, 2 Corinthians 11:5).