How can I get notified when there's an incomplete payment?

How am I supposed to know that its an incomplete payment? This seems like
a huge flaw in your system? Its not only time-intensive to have to worry
about this and reminding myself to check for it but more importantly
embarrassing to have to go back to a client and ask if it went through on
their end. Its a very cumbersome unsophisticated system, especially if I'm
not even notified that I have to log in to click a button. Is there any
way to improve this or prevent it.

We already have this feature available, you can get notified when there's an incomplete payment on your form, an email is sent notifying that you have received a submissions without a payment done. 

Here is the guide that will allow you to get the notification about an incomplete payment: How-to-Setup-Incomplete-Payment-Notification-E-Mail

When you log into your account and check the submissions you will find the submissions in the "Incomplete Payments" section, this guide will help you to view your them: How-to-view-incomplete-payments

So you will not have to ask or check constantly your submissions as you get notified instantly when there's an incomplete payment. 

Hope this helps.