When posting a form, if you use ‘name’ as an input name then there seems to be a problem with submitting.
Try changing:
<input type="text" name="name" class="form-contact__text" />
to:
<input type="text" name="the_name" class="form-contact__text" />
Also, best to give your submit button a more unique name in case it conflicts with any plugins/themes – then check for the submission of that in your process_contact_form()
function.