TL;DR - For me, it helped removing "Gaming Services" from my PC.

Many users have reported that Max Payne 3 randomly pauses every few seconds.
Instead of fixing their bad code, Rockstar recommends a bunch workarounds that didn't work out for me:
https://support.rockstargames.com/de/articles/200152866/Pausing-to-the-Pause-screen-randomly-in-Max-Payne-3-for-PC

Like other Rockstar games, Max Payne 3 checks if your input device is still pluged in.
But unlike the other R* Titles, this game also checks for more than just the input devices.
This means that a faulty device could cause your game to pause, even if it's not an USB-Device.

If this happens to you, do the following steps:

First you have to find out what is causing the bug.

1. Open the Windows Event Viewer:
   ⊞ Win + R type eventvwr
   
   
2. Start the game and wait for the next pause.
   
   
3. After the bug occurs, immediately Alt + Tab out and check the Logs under Windows Logs > System .

In my case, this was the warning:
Disk 9 has been surprise removed.

(If you have a different warning, try to find out which device causes the issue and disable it in the Device Manager.)

So my PC only has 5 physical hard drives.
After a bit of research it turned out, that Disk 9 is a virtual "xvd disk", which is used by Microsoft for the Game Pass and the Gaming Services.

I tried disabling all Microsoft Gaming Services but that didn't help. The only fix was to uninstall the Gaming Services.
This can't be done via the settings menu, but with a PowerShell command:
https://answers.microsoft.com/en-us/windows/forum/windows_10-windows_store/gamingservices-and-gamingservicesnet/04e2fc02-ff76-444b-8ed3-d03a8bd55e3c

Please let me know if it helped, or if you've found a better way to fix this.