The fact that you have just put a raster image in your SVG isn't the actual reason for what you are seeing.
All it means is that when the scaling of the SVG works properly (see below), you won't see the benefits of using vector artwork. When you scale up vector artwork, you don't get the "jaggies" (blockiness) that you get when scaling up bitmaps. If your SVG just contains a bitmap, it is pretty much the same as just using the bitmap.
The actual problem here is that Inkscape doesn't include a viewBox attribute in the SVGs it saves.
When you remove the "width" and "height" attributes, they default to "100%". Which tells the browser to scale the SVG to fit the parent container. Unfortunately for this scaling to work, the browser needs to know what the dimensions of the SVG content are. That is what the viewBox attribute is for.
Illustrator adds the viewBox attribute, Inkscape does not.
To see what I mean, add the following to your <svg>
tag after removing width and height:
viewBox="0 0 319 177"
You should find that your image is no longer clipped, and will resize when the page is resized.