A powerful caution is given against the dangers of assimilation and the gradual decline into idolatry. The laws of the Torah are designed to create a protective boundary, ensuring the Israelites do not intermarry or blend in with the surrounding nations [מצודת דוד, שטיינזלץ, מלבי״ם].
The foreign populations remaining in the land fall into two distinct groups: those who were simply left behind by chance after the conquest, and those God purposefully allowed to remain as a test for the Israelites [מלבי״ם]. Regardless of why they are there, these surrounding nations hold no true importance on their own [שטיינזלץ].
As part of maintaining this separation, there is a strict warning against interacting with the false gods of these groups. This restriction goes beyond a personal ban on taking an oath in the name of an idol. While [רד״ק] understands this as a rule against forcing a fellow Israelite to swear by false gods, other commentators maintain that it forbids demanding even the foreigners themselves to take an oath by their own idols [מצודת דוד, חומת אנך, שטיינזלץ].
Ultimately, this sequence of warnings outlines a clear and tragic chain reaction where one sin inevitably leads to another [חומת אנך, מלבי״ם]. Assimilation and intermarriage act as the gateway to increasingly severe offenses. The descent begins simply by mentioning the names of foreign gods. From there, it escalates to swearing by them, which leads downward into actively serving them. The final and most severe step is physically bowing down to these idols, an act so grave that it carries the death penalty. Therefore, strictly avoiding the first, seemingly harmless steps is the only way to guarantee protection from the absolute worst offenses of idolatry.