Now that we know the board type (Zero) — which took a couple of questions to elicit — it is possible to give a reasonable answer. Had it been a Uno I would have said that no, it doesn't do that, and it would take too much program space to implement it.
Is there another option to have an existing program erased or replaced with something benign ...
No, not that I am aware of, unless the documentation for the Zero says otherwise.
If not, what is the level of difficulty to modify the standard bootloader to implement this option?
Moderate, I would guess.
As for the Zero, with 256 KB of programmable memory it probably could be implemented by changing the bootloader. Any such change will, of course, reduce the amount of memory available for other tasks, but perhaps that isn't as big a deal if you start with 256 KB.
You would have to detect the triple-click somehow, for code running right after reset this could be tricky, but possibly achievable, as you say it has been for double-clicks.
Find the source code for the bootloader, see how it detects double-clicks, make it detect triple clicks, and have your "benign" program stored in the bootloader as a block of data ready to be copied into program memory.
There are a few alternatives you could consider:
If you have a spare pin in your project, make it so that if (say) pin 13 is grounded at startup, do nothing (other than go into a tight loop). This could be done before anything that might make reprogramming hard, like going into low power mode.
Reprogram it using SWD programming (rather than the bootloader). I don't know the exact details for the Zero, as I don't own one, but there must be a way, otherwise you wouldn't be able to put the bootloader onto it in the first place. Judging by what I read elsewhere the Atmel ICE is the device you need to reprogram it in that way.
Judging by what I read here you can access the bootloader by selecting Arduino Zero (programming port) as the board, and Atmel-EDBG as the programmer. Then you could tell the Atmel-EDBG to reprogram the board. My details here are hazy but you could look into that as an option.