I think the question I'd ask before rooting is what you plan to do with root?
If this is a hardware problem, such as a broken/stuck button, then no software modification will fix it. Rooting by itself won't change anything, but it might give you more options for working around a broken button if you choose not to get it fixed. However, as you've seen booting while pressing a button can put you into special modes, and as that happens before Android is started, modifying the android ROM (which is what root does) won't change that behaviour, in which case my guess is that you'll still end up in safe mode.
So it's your phone, but if this is a hardware fault (and if a reset didn't fix it then that's the most likely thing) then I'd personally get it fixed first and worry about root later. To be sure there are no debates about your warranty you'd want to unroot and return to stock before sending it in, so why go to the hassle of rooting only to unroot?
If you do want to root, Scotty has written a detailed guide to our normal rooting methods: The HTC One root thread
. You won't be able to use the exact method I used because if you are on 4.2.2 already then the tool I used (revone) will no longer work, but the HTCDev (official) unlock described in Scotty's thread still does. However, if you do that they will be able to tell that the phone has been modified.
As for the tool you link, it's interesting but the description of what it does isn't very detailed. My guess is that it's probably easier to undo completely than the official unlock, but is a much more limited form of rooting than we'd normally do. I say that because a key tool for rooters is a custom recovery, which let's you modify or replace ROMs and, most importantly, take full system backups, and from what I can find on the tool provider's site I suspect that that tool does not install one, just modifies the current ROM to add root to it.
Bottom line, if that tool works to root it will probably work to unroot, so you could probably use it to investigate whether it actually helps. But no promises, because I've never seen that tool before, and if you were interested in rooting for the reasons most are then I think I'd recommend using other methods instead. I certainly wouldn't root without a plan for what you want to do afterwards, because root by itself won't fix anything, just gives you the option to do other things you couldn't otherwise have done.
Sorry if that all sounds a bit negative. It's just that much as I am a fan of rooting myself, I'm reluctant to recommend it if I suspect the problem is hardware.