Since Linux Mint probably uses a fairly recent version of GRUB, its core has probably been embedded into the MBR and the disk blocks between the MBR block and the beginning of the first partition. Its configuration and possibly some modules are probably loaded from the
/boot/grub directory on
sda5. The fact that GRUB is starting and allowing you to choose between FreeDOS and Mint even though the FreeDOS partition is marked active supports this.
I'm not sure, but I suspect that GRUB is probably starting FreeDOS by chainloading the first block of the
sda2 partition, so the bootloader of FreeDOS is probably intact. You shoud read the FreeDOS boot section in
/boot/grub/grub.cfg in Linux Mint to confirm this.
If that's true, the only thing you'll need to do in order to effectively remove GRUB from your system, is to replace the GRUB boot code in the MBR with a more traditional MBR boot code. The most straightforward tool for this task is probably ms-sys. You can probably find it as an optional package for Mint, so you don't have to build it from source code.
Warning: either of the following commands will instantly replace the boot code in the MBR of the specific disk (without harming the partition table), overwriting GRUB and making your Linux Mint unbootable without an external boot media. Undoing this operation will require reinstalling GRUB to the MBR, i.e.
sudo grub-install /dev/sda or similar. Make sure you really understand what you're doing before proceeding.
The most "standard" MBR boot code would probably be the one that's used by Windows XP:
sudo ms-sys -m /dev/sda
But if you want the system to be absolutely free of Microsoft's intellectual property, you might want to use the public domain MBR boot code from the SYSLINUX bootloader project instead:
sudo ms-sys -s /dev/sda
Technically, there will be some left-over bits of GRUB in the disk blocks between the MBR and the beginning of the first partition, but once the MBR part of GRUB has been overwritten with a more traditional boot code, those leftover parts of GRUB will not be accessed by anything. A careful forensic examination might reveal that GRUB had once been installed to this system, but that's about it.
I'm not very familiar with the current state of FreeDOS, but if it still does what classic MS-DOS did, the command for overwriting GRUB in FreeDOS might be