Magento 2 – Country Name Not Showing in Billing/Shipping Address at Checkout

You can set separate addresses by creating another address in the customer section and set that to default shipping address

As Aaron pointed out the form is added in Magento/Checkout/Block/Checkout/LayoutProcessor.php. With this information I developed a module with an after plugin that hooks onto that processor:

app/code/<vendor>/<module>/Model/Checkout/LayoutProcessorPlugin.php

<?php

namespace <vendor>\ReorderBillingForm\Model\Checkout;

class LayoutProcessorPlugin
{

    /**
     * @param \Magento\Checkout\Block\Checkout\LayoutProcessor $subject
     * @param array $jsLayout
     * @return array
     */

    public function afterProcess(
        \Magento\Checkout\Block\Checkout\LayoutProcessor $subject,
        array $jsLayout
    )
    {
        // get billing address form at billing step
        $billingAddressForm = $jsLayout['components']['checkout']['children']['steps']['children']['billing-step']['children']['payment']['children']['afterMethods']['children']['billing-address-form'];

        // move address form to shipping step
        $jsLayout['components']['checkout']['children']['steps']['children']['shipping-step']['children']['billing-address-form'] = $billingAddressForm;

        // remove form from billing step
        unset($jsLayout['components']['checkout']['children']['steps']['children']['billing-step']['children']['payment']['children']['afterMethods']['children']['billing-address-form']);

        return $jsLayout;
    }
}

app/code/<vendor>/<module>/etc/module.xml

<?xml version="1.0" ?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:Module/etc/module.xsd">
    <module name="<vendor>_<module>" setup_version="1.0.0"/>
</config>

app/code/<vendor>/<module>/etc/di.xml

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
        xsi:noNamespaceSchemaLocation="urn:magento:framework:ObjectManager/etc/config.xsd">
    <type name="Magento\Checkout\Block\Checkout\LayoutProcessor">
        <plugin name="reorder-billing-form"
                type="<vendor>\<module>\Model\Checkout\LayoutProcessorPlugin" sortOrder="<yourOrder>"/>
    </type>
</config>

app/code/<vendor>/<module>/registration.php

<?php

\Magento\Framework\Component\ComponentRegistrar::register(
    \Magento\Framework\Component\ComponentRegistrar::MODULE,
    '<vendor>_<module>',
    __DIR__
);

This successfully reorders the billing address form (and hopefully saves some headaches for other people). But there is still work needed on the javascript(?) that handles the billing address is the same as shipping address mechanism. As this still works the "standard" way.

Additional info:

I saw that in the backend if you create a new order the layout is exactly as wanted. The billing form is "before" the shipping form and the logic is the other way around too. If I can find the time I think it might be beneficial to look at the code there. Maybe it is possible to use it in frontend too.