$content_width is used to set the default width of embeds, for situations when an element needed to specify a width. This is how WP knew the width of the main content area.
However, this was back before the push for responsive design, and modern CSS with fluid embeds and responsive breakpoints. So it isn’t needed much but it can still result in odd embeds if you set a bad value. Content width sometimes gets used in other edge cases too, but that was the main use case and its reason for being.
So I suggest:
- Figure out the content width on the most common desktop resolution, and use it to set
$content_widthinfunctions.php - Then, apply CSS to adjust embeds responsively/fluidly as you would normally do