The context here has to do with David's lust for more women.
Not quite. The context has to do with sending someone to their death, so that one could then proceed to marry their widowed wife.
Just as anything ranging from simple theft to aggravated robbery is forbidden by the Mosaic Law, but the pursuit of wealth is not (provided that the poor, the widows, and the orphans are not forgotten, which constitutes a major theme in the prophetic books), so also adultery (in all its violent and non-violent forms) is prohibited as well, but the pursuit of (preferably virgin) wives (and concubines) is not. Of course, Christ does indeed explicitly and repeatedly condemn the acquiring of wealth in the Gospels, but He lived one and a half millennia after Moses.
I'm struggling with a verse that seemingly supports God's approval of concubinage and taking of many wives. [...] This makes me feel uneasy leaving this lingering in my mind.
Does God's tacit or implicit tolerance of polygamy make you feel uneasier than Him explicitly commanding that various types of (sinful) people be put to death by stoning, or that entire cities, including their (male) children, be blotted out of existence ? If you accept the latter as biblical fact, why not the former as well ?
Or perhaps you are simply bothered by the fact that the Gospels contain no polygamy-related equivalent of John's Pericope Adulterae, explicitly condemning the practice ?
Is to be interpreted as approval of polygamy or is the main focus something else ?
As already mentioned above, the Mosaic Covenant does not forbid polygamy (Moses himself, through whom the Law was given, had multiple wives, as explicitly mentioned in Scripture); were that not the case, then there would have been no logical reason for the Prophet Nathan not to explicitly mention this to David as well, just as he explicitly mentioned his (indirect) murder of Uriah (in a manner not explicitly forbidden by Mosaic Law; after all, it's hardly a sin for a king to send his most trusted and valiant fighter into battle) so as to legally take his widowed wife as his own, since adultery, unlike David's conniving plot, was explicitly forbidden by the Law.
As the Pharisees after him, David kept the letter of the Law, while breaking its spirit, by infringing upon its intended purpose and meaning.
As Christ in New Covenant times, Nathan the Prophet is tasked by God to remind him of that.
Is to be interpreted as approval of polygamy or is the main focus something else ?
The main focus is (obviously) obedience to God's law, and trust in His divine providence.
Did he desire (yet) a(nother) wife ? If so, then why not acquire one by then-legal means, having faith that the same God that delivered all other blessings into his hands will not fail him this time either ?