The company I work for has a standard attendance policy. The problem is not with the policy itself, but with the enforcement from the management.
Too often, our management wants to use attendance as a way of coaching bad employees out of the company, but they end up jamming themselves up because they don't enforce the policy across the board and end up with a human resources issue of unfair treatment. They want to write up Billy every time he's three minutes late because he also happens to be a lazy employee (which, incidentally, also happens to be the core reason you're actually performance managing him, right? )
, but the cute girl Julie comes in late, winks, smiles and goes on about her business without so much as a talking to.
My advice is to start enforcing the policy you have in place, but be willing to enforce it across the board and lose a few employees. I will caution you though, often the employee that you have a good relationship with will think that the new enforcement won't apply to them because of the good relationship they have with you. It may be challenging to write up/suspend/fire that person, but if they won't give you the respect of showing up to work on time, how good of an employee are they for you, really?
Also, your view of the pay issue is blurred in my opinion, your mindset is that the lazy people are getting paid the same as the productive employees so why perform? My answer to that would be, How did the team get the impression that you will tolerate people being late time and time again with no reprimands? Are the inmates running the asylum, or are you the one in charge?
I'm not sure what type of business you're running, but I'm sure your employees are not volunteers, and I am sure they are being paid to show up to work, and do a job. The people above you may dictate the pay, but you do have the ability to dictate who gets to keep their job or not, and unless you're running a soup kitchen, your employees are being paid, so when they come in late, they're not wasting their time, they're wasting your time.