You still had parts of it calling my old script (#budget_range).
I changed it all up for what I think you're looking for. Here you go :)
var year = 1897;
var min_year = 1897;
var max_year = 1996;
html = '<input type="range" name="birth-year" step="1" class="wpcf7-form-control wpcf7-range wpcf7-validates-as-number" aria-invalid="false" min="'+min_year+'" max="' + max_year + '" value="'+year+'" name="year" id="year_range" />\
<span>' + year + '</span>';$(".birth-year").append(html);
$('#year_range').on("change mousemove", function() {
$(this).next().html($(this).val());
});
Hope this helps!