I think the proposition: "the work of the Lord" = "evangelism"--is a little narrow in its application. If we take the "overview" as if it was defending that proposition, here's what I have in response:
--It seems to put evangelism right at the tip-top of every to-do list for every believer all the time; the argument as it is put forward levels every recipient and every reader of Paul's letter to a single office, ignoring even the obvious fact that Paul was an apostle and there are no more living in this world. The church has a ministry (pastors, elders, deacons), to which has been delivered the continuance of the service begun by the apostles; Timothy in 1Cor.16:10 is an ordained minister (by the laying on of hands, 1Tim.4:14; 2Tim.1:6) so "work of the Lord" is broadly applicable to his Christian calling. The same can be said of Stephanus and the other men mentioned in the context of 1Cor.16:10-15; again, the indiscriminate application of Paul's commendation to literally any "saint" goes beyond the apostle's intent for his words. Likewise, Php.2:30 has special reference to the work of Epaphroditus, who was commissioned (if not an ordained man) for delivery of a church-message and gift to Paul. Eph.4:12, while the modern take on "everyone a minister" here is a popular interpretation, should be understood narrowly as the "work of the ministry," i.e. those who conduct the official government labor of the King.
--The idea that the word for "labor" is in any sense a "technical term" (even when coupled with earlier, "work of the Lord") having a specific ministry-sense when used in the NT assumes what must be proved for effect; it is a rather ordinary Gk word, meaning toil or weary-work, which can describe the unceasing demands of ministry; but Christians of all sorts are called to diligence in their God-honoring activities, both inside and outside of the church or covenant-community of faith, which diligence often leads to weariness; if Jesus could be so weary from a journey (Jn4:6) then anyone can get so weary regardless of the nature of the trip.
--Mt.11:28 uses the verb found in 1Cor15:10 to describe the weariness of unbelievers; Act.20:35, Paul uses the term to describe his self-support that fell outside of his direct ministerial labor; arguing a conceptual parallel between vv10 & 58 again seems to assume what needs to be proved.
It's possible the article you reference does not intend to limit the phrase in question to evangelism, but to be sure to include evangelism in every of description "work of the Lord." That is less objectionable, but it still may arise from a modern, western, egalitarian opinion about the true composition and conduct of the church through its various members' activity. Paul is getting close to his final words in the letter he writes to Corinth. He has addressed many issues, and directed his comments to both leaders and followers in the church, to people who are behaving well and to people in need of discipline in one form or another. Certainly, for those who are called to the work, and those who are able to evangelize, the description "work of the Lord" would include that specific demand.
It seems quite reasonable to read 1Cor.15:58 as language that applies to everyone in the church; but in the nature of the case, not commending one single thing to everyone by the general terms of it. "The work of the Lord" may be both broadly and narrowly conceived. I'm not sure where Hodge goes to connect that first possible referent (putting sin to death), though the idea would plainly be tied to the final termination of sin (v56) at the resurrection and Judgment Day; but I suppose him more inclined to the second option: the different works that fall to the range of believers' service rendered to the Lord in the midst of a variety of callings. It is all bound together in the single concept: "the Lord's work," which we all should do "heartily, as unto the Lord," Col.3:23, because it all falls under his purview, and under his unified sovereign disposition. Evangelism is one part--important, vital, indispensable--but not the only part of the work of the Lord.
Finally, evangelism (telling unbelievers the gospel) and gospel proclamation (in the church, typically to presumed believers) could be thought of as the principal goal of the church and the Christians within. With that in mind, we sometimes speak of the pastor (and others) doing "gospel-labor," without limiting that labor or watering down the word "gospel." We simply mean that in some mutual way, everything the church is a part of or is incorporated into the church's business ends up in service to the main thing: Christ, and him crucified (i.e. the gospel-in-a-nutshell). I don't think Hodge would have a problem with that, or with combining it with what he says about everyone's God-oriented callings.
Paul's imperative neither 1) summons every hearer to the exact same service, let alone to imitation of his apostolic ministry; nor 2) is it narrowly applied to just that class of post-apostolic service that does have some duty to carry on the residue of ministry in the church. The direction is broad enough to cover everyone in his calling, especially the general duty to love as Christians (one of this epistle's great teaching subjects, 1Cor.13); surely all our efforts to fulfill that ideal qualifies as "the work of the Lord," does it not? All the commandments of the Lord summed up in one word: Love. (Rom.13:10; Gal.5:14)
