What really happens on modern PC hardware booted in 16-bit legacy BIOS MBR mode when you store a byte such as \'1\'
(0x31) into the VGA text (mode 03) fram
Reading through various modern Intel CPU and Platform Controller Hub (PCH) datasheets, it doesn't appear that the necessary hardware is implemented. There doesn't seem to be any way to generate an SMI (System Management Interrupt) in response to processor accesses of the VGA frame buffer (physical addresses 0xA0000 - 0xBFFFF).
The memory controller in the CPU will either route accesses to VGA frame buffer to the integrated graphics controller, the PCI Express port connected directly to the CPU, or the DMI interface connecting the CPU to the PCH. While it's possible route parts VGA frame buffer separately, this appears only meant to support a separate MDA (Monochrome Display Adapter) device. The integrated graphics controller is not well documented so it's possible that it can be configured to generate an SMI on VGA frame buffer accesses, but this seems unlikely. In any case, it wouldn't work with discrete graphics.
Intel PCH's also don't seem to have any support for generating SMIs in response to VGA frame buffer accesses. This would be the most natural place for it, as it already has support for generating SMIs in response to I/O accesses to the keyboard controller, IDE controller and other legacy devices. It possible that there's some undocumented feature that does this, but it's not included in the lists of possible SMI sources given in the PCH datasheets.
Theoretically, it would be possible for a motherboard manufacture to connect a fake VGA device to the PCH through a PCI Express port and then generate SMIs using a PCH GPIO pin. However, I'm not sure this will work in practice. By the time the CPU gets the SMI it could have moved on to executing other instructions and it wouldn't be possible to examine the CPU state at the time of the frame buffer access.
(A similar problem happened with SoundBlaster 16 emulation on the SoundBlaster Live. It would generate a PCI SERR# when the legacy SoundBlaster ports were accessed, which would generate a NMI on the CPU. Unfortunately the emulation would break on many Pentium 4 motherboards because the NMI would arrive on the next or subsequent instruction.)