3 This (i.e. 'praying to God for rulers so that Christians may go about life peacefully') is good, and it is pleasing in the sight of God our Savior, 4 who desires all people to be saved and to come to the knowledge of the truth. 5 For there is one God, and there is one mediator between God and men, the man Christ Jesus, 6 who gave himself as a ransom for all, . . . (1Tim 2:3-6 ESV)
This passage agrees with your point. 'God desires all people (lit. "men") to be saved.' (1Tim 2:4) And there is a covenant for making that possible. The covenant is between "God" and "men" (same word for "people" above). And the "mediator" is Jesus. The only plausible covenant being spoken about would be the New Covenant.
Note the contrast between "men" (or "people") which is without the article, and "the man" (Christ Jesus). If "men" or "people" were referring to a very select group one would expect an articular "men."
The NWT wiggles around this problem by rendering "all people" as "all sorts of people." It is an anti-Christian rendering in that it denies a very large portion of Jesus' ransom. It could possibly be classed with 1Jo 2:22a.