Without the extra VRAM, it takes hundreds of times divided by batch size longer due to swapping, or tens of times longer consistently if you run the rest of the model on the CPU.
But if you just want to double the memory without increasing the total memory bandwidth, isn't it a good deal simpler? What's 1 more bit on the address bus for a 256 bit bus?
Instead, it appears entirely possible to double VRAM size (starting from current amounts) while keeping the bus width and bandwidth the same (cf. 4060 Ti 8GB vs. 4060 Ti 16GB). And, since that bandwidth is already much higher than system RAM (e.g. 128-bit GDDR6 at 288 GB/s vs DDR5 at 32-64 GB/s), it seems very useful to do so, though I'd imagine games wouldn't benefit as much as compute would.
You can see this with overclocking VRAM. Greatly benefits mining, slightly or even negatively benefits gaming workloads.
This extends to system RAM too, most applications will see more benefit from better access times rather than higher MT/s.