Intercooler is all about getting HTML into your DOM that you get from AJAX requests without actually writing any Javascript. Alpine is more about Interactivity within the current page, without doing a request first. It keeps the state in a data object.

Of course you can do similar things as you would do in Intercooler, but it would be explicit in the Javascript part. See this example they give for prefetching dropdown content from the server

    <div x-data="{ open: false }">
        <button
            @mouseenter.once="
                fetch('/dropdown-partial.html')
                    .then(response => response.text())
                    .then(html => { $refs.dropdown.innerHTML = html })
            "
            @click="open = true"
        >Show Dropdown</button>

        <div x-ref="dropdown" x-show="open" @click.away="open = false">
            Loading Spinner...
        </div>
    </div>

Having used Intercooler a few times now, it feels like the worst of all worlds since you now have to write and return HTML at AJAX endpoints and keep that HTML synced with the rest of your HTML (e.g. consider CSS) instead of just returning data from an API endpoint.

You end up with a bunch of made-for-Intercooler endpoints instead of thoughtful, general, data API endpoints.