The headword, here in the dative plural, is the aorist active participle of a compound of tugxa/nw (tau 1147, epsilon 3344), e)pitugxa/nw, used in a variety of senses (cf. epsilon 2738, epsilon 2674, epsilon 2736, epsilon 2673, kappa 1009) over the long development of the concept of tyche from success to mere accident (tau 1232, tau 1233, tau 1234, eta 286 and refs.). It may be used of someone accurately hitting a target (with a bit of luck), all the way to the sort of person you happen to run into on the street (as glossed here); cf. LSJ II.4, web address 1.
[1] The author of this quotation -- apparently in reported speech -- is unknown. Adler’s suggestion of Polybius is implausible, as he does not use e)pituxw/n in the sense required here (an observation I owe to F.W. Walbank in correspondence). If the passage refers to electing consuls carefully at Rome in times of danger, it might come from any one of many sources. Possibly, though, it refers to the choice of rejecting consuls in favour of the Roman institution of dictatorship (delta 1110, delta 1111, delta 1112; OCD(4) 448; cf. early dictators mentioned at iota 522, alpha 4696, phi 627, chi 240).
Either way, the use of e)pitu/xousi suggests Appian, who uses the participle quite often for those who arrive at high status without deserving it, a more likely sense than the one given by the gloss. They would then be in sharp contrast with L. Quinctius Cincinnatus (*kikinna/tos or *kigkinna/tos: kappa 2732, lambda 846, OCD(4) 1250), dictator in 458 BCE against the Aequi, and chosen partly because he cared nothing for status. The quotation might equally refer to Manlius (mu 105), who actively sought a dictatorship.
Another possible source is Plutarch, for he uses toi=s e)pituxou=si in this sense in a similar context at Precepts on administering a republic 801b1.