Embedded content is not responsive “by magic” someone needs to write code to actually make it so, and some default needs to be there in any case, and actually not all site are responsive, and some even can not be responsive.
You might want to establish a base line, for example the size of the image being displayed when the page is loaded before responsiveness kick in, to make it a better experience for the majority of your users (95% of them are on desktop? maybe good idea to start with a desktop mode).
OTOH, regardless of responsiveness, the $content_width is a global and therefor it is better to avoid using it and pretend it does not exist as much as you can.