I used to get code back from developers EACH AND EVERY TIME with massive bugs like: unable to register, unable to login, unable to add content. I wrongly assumed that they at least ran through and checked for any bugs they introduced before sending me the new code. So each and every time I got code back I had to go through manually and check it, sign in, log out, register, add content, delete content, edit content, add category, etc...
I wish someone could make a simple service that allows me to set up my web app, set up test parameters that it tests each and every time, and tell me if it failed or not. I want to automate my babysitting.
There is some reason that your developers aren't engaged in the work. Figure out why they don't care about working code or the user experience and fix that.
If you plug the obvious holes, you won't have fixed your quality problems; you'll just shift them to the places where you won't notice them right away.
I can't agree more with wpietri above. There surely is a non-technical problem at play. That isn't to say that you shouldn't try to automate your babysitting, but if your need for babysitting is that severe, you have other problems.
Selenium is not the prettiest tool out there, but as the homepage says, Selenium automates browsers, making it ideal for running tests that you describe:
I wish someone could make a simple service that allows me to set up my web app, set up test parameters that it tests each and every time, and tell me if it failed or not. I want to automate my babysitting.