Is there a way to select multiple shapes and turn them into Bezier paths at once?

You can try this script:

- select your shape paths and run the script

convertToBezierPath.jsx

function convertToBezierPath(i) {
// DeselectAll
	app.executeCommand(2004);
    selectedProperties[i].selected = true;
// ConvertToBezierPath 
    app.executeCommand(4162);
    i++;
    if (i < selectedProperties.length) {
        $.sleep(500);
        convertToBezierPath(i);
    }
}
var proj = app.project;
var thisComp = proj.activeItem;
var selectedProperties = thisComp.selectedProperties;
var i = 0;

convertToBezierPath(i);

if the $.sleep() value is too low, you may have problems, in my case, if I set the value to 100, it returns this:

If you create the shape by clicking and dragging, look at the toolbar and choose the Bezier Path option. Then, no conversion is required. If you need precise dimensions, the only option would be to convert the Paths to Bezier Shapes one at a time or try writing or looking for a script.

Another option would be to create you simple shapes (no gradients or brushes) in Illustrator, import the AI file as a composition, then convert the vector layers to shape layers.