Hello and congrats on completing this project! It looks great! I wanted to mention a few things.
To answer your question about the button. I don't think it is an issue since I have also used a button for form submissions. For my example, I used <button type="submit"...> to use a button to submit forms.
Another thing I wanted to mention is <form action=' '> must not be empty since it causes html issues. Instead of leaving it empty, use the '/' to fill the action since that won't do anything when the form is submitted.
When using <label for....> to label inputs, make sure the for="..." matches the ID of the input. For example <label for="sub"> <input id="sub" > would be considered good coding practice.
Finally, the last thing I wanted to mention is for your terms and service html structure, you can get rid of the <p> and </p> elements since it's not allowed inside the <sub> element. One alternative is to get rid of the <sub> element and leave the <p> element as is.
I am still currently learning more about screenreaders and how to improve its accessibility so I am not quite sure why the screenreader is only reading the terms and conditions paragraph. I do wish you best of luck on how to solve that issue!