Can’t get a JSON object in response to an Ajax request with wp_ajax

I have a problem with WordPress and Ajax.

This is my JavaScript part (I trimmed it a bit):

var posts = $.ajax({
    type: 'POST',
    url: ajaxurl,
    async: false,
    dataType: 'json',
    data: { action: 'myAjaxFunc' },
    done: function(response) {
        return response;
    }
}).responseText;

$.each(posts, function() {
    $('#someSelect').append( $('<option</option>').text(this.name).val(this.id) );
});

My PHP code is as follows:

function myAjaxFunc() {

    $posts = get_posts( array(
        'posts_per_page'   => -1,
        'orderby'          => 'title',
        'order'            => 'ASC',
        'post_type'        => 'my-post-type',
        'post_status'      => array( 'publish', 'draft' )
    ) );

    $list = array();
    foreach ( $posts as $post ) {
        $list[] = array(
            'id'   => $post->ID,
            'name' => $post->post_title,
            'link' => get_permalink( $post->ID ),
        );
    }

    header("Content-type: application/json");
    echo json_encode( $list );
    die;
}
add_action( 'wp_ajax_nopriv_myAjaxFunc', 'myAjaxFunc' );
add_action( 'wp_ajax_myAjaxFunc', 'myAjaxFunc' );

The script gets the Ajax response from admin-ajax. Unfortunately the console throws an error when it gets to the each statement in the JavaScript code… it says:

"Uncaught TypeError: Cannot use 'in' operator to search for '4' in Array".

If I do a console.log of my “posts” var I get a string ‘Array’. No matter how I pass the $list variable in PHP it will always return a string. The query returns posts elsewhere, so it’s not empty. I tried without json_encode, with and without declaring header, using wp_send_json(), putting ob_clean() before echoing the array, putting the array into an array… But it always gets into ajax as a string Array and each cannot cycle through it.

This should be a very simple thing and I can’t understand why it’s not working. I don’t have other JavaScript or PHP errors or warnings and everything else runs fine.

BODA82’s answer helped, but eventually I realized that I should have replaced responseText with responseJSON method in my JavaScript code. In the example below I was storing the Ajax response results in a variable. I didn’t know there was a specific method to get the response in JSON. In a such way the object/array with get_posts() results is returned correctly and not as a string:

posts = $.ajax({
    type: 'GET',
    url: ajaxurl,
    async: false,
    dataType: 'json',
    data: { action : 'getHotelsList' },
    done: function(results) {
        // Uhm, maybe I don't even need this?
        JSON.parse(results);
        return results;
    },
    fail: function( jqXHR, textStatus, errorThrown ) {
        console.log( 'Could not get posts, server response: ' + textStatus + ': ' + errorThrown );
    }
   }).responseJSON; // <-- this instead of .responseText

Note to self, but also general advice: if you can’t fix something in the evening it’s a sign you should go to bed, read a book, and count stars. An answer will be found the next morning, the earlier the better 😀