Interestingly, this passage (or hymn) was used by the apostle with reference to how Christians relate to each other, in particular (see context, vv.1-4, 12-16).  ‘Incarnational’ ministry is not so much about ‘cultural engagement’ per se as it is about loving others, and our fellow Christians especially.  This same logic appears in Paul’s arguments regarding eating meat sacrificed to idols in 1Corinthians 8-9, in which Paul is concerned about ’scandelzing’ the weaker brother’s conscience, and embolding  them to sin against it (8:4-13).  He ends his argument with the famous summarization of his ministry: “Though I am free from all men, I have made myself a slave to all, so that I may win more.  To the Jews I became as a Jew…,” etc.  That is, the ‘incarnational principle’ is all about our self-sacrificial love for others, whether Jews, Greeks, or the church of God (see 1Co.10:32).  It is a call to self-control, as you stated, in contrast to the self-indulgence of Israel in the wilderness (10:6ff.) - a call to run the race with discipline and diligence (9:24-27).  It is a call to sacrifice our liberty for the sake of salvation (ours and that of the other). 

As articulated in 9:19-23, it is a ‘missional’ strategy of the apostle - his ‘incarnational’ approach in ministry.  Yet Paul cites this as an example of a broader principle of sacrificing our ‘rights’ for the sake of the gospel (e.g., 9:1-15) - a theme with strong resonance in Philippians 2.  Within the broader context of 1Corinthians, this is an instance of what it means that we “have love” (13:1-13).  And this love is ‘incarnated’, if you will, in our pursuit of the ‘edification’ of the other (cf. 10:23-24; 14:1ff.), whether the up-building of our brothers (which is diametrically opposed to placing ‘stumbling blocks’ in their path) or seeking the salvation of the lost.   That is to say, the gospel shapes what our love will look like.  In fact, the gospel so shapes the apostle’s life and mission that his very approach in ministry is an embodiement of it.  He doesn’t just preach it; he lives the gospel.  Perhaps this is what Paul means when he writes, “I do all things for the sake of the gospel, so that I may become a fellow partaker of it,” 9:23? 

Nevertheless, salvation through Christ (the gospel) is the theological ’backdrop’ of this entire discussion.  Paul is concerned that our lack of love for the other, which calls us to self-sacrifice (of even legitimate freedoms/rights) will result in the ‘destruction’ of the weaker brother (8:11), and on the other hand, that true love leads us to sacrifice freedom in order that “by all means” we might “save some” (9:22) - behavior which he calls us to emulate (10:33-11:1). 

So there is an “in order that” in love: that others might be blessed, edified, built-up, etc.; and nothing short of salvation can be the aim of true love. 

Have you read Stott address these issues in ”Christian Mission in the Modern World”?  He addersses these topics in a very balanced and thoughtful way.