How do you set a default root object for subdirectories for a statically hosted website on Cloudfront?

There IS a way to do this. Instead of pointing it to your bucket by selecting it in the dropdown (www.example.com.s3.amazonaws.com), point it to the static domain of your bucket (eg. www.example.com.s3-website-us-west-2.amazonaws.com):

Thanks to This AWS Forum thread

Activating S3 hosting means you have to open the bucket to the world. In my case, I needed to keep the bucket private and use the origin access identity functionality to restrict access to Cloudfront only. Like @Juissi suggested, a Lambda function can fix the redirects:

'use strict';

/**
 * Redirects URLs to default document. Examples:
 *
 * /blog            -> /blog/index.html
 * /blog/july/      -> /blog/july/index.html
 * /blog/header.png -> /blog/header.png
 *
 */

let defaultDocument = 'index.html';

exports.handler = (event, context, callback) => {
    const request = event.Records[0].cf.request;

    if(request.uri != "/") {
        let paths = request.uri.split('/');
        let lastPath = paths[paths.length - 1];
        let isFile = lastPath.split('.').length > 1;

        if(!isFile) {
            if(lastPath != "") {
                request.uri += "/";
            }

            request.uri += defaultDocument;
        }

        console.log(request.uri);
    }

    callback(null, request);
};

After you publish your function, go to your cloudfront distribution in the AWS console. Go to Behaviors, then chooseOrigin Request under Lambda Function Associations, and finally paste the ARN to your new function.

There is one other way to get a default file served in a subdirectory, like example.com/subdir/. You can actually (programatically) store a file with the key subdir/ in the bucket. This file will not show up in the S3 management console, but it actually exists, and CloudFront will serve it.