A way to rephrase the "because this is the way the founder did it" answer is that the founder trained in a way that didn't requite a tsuba. This training was formalised and put down as a methodology by Saito Morihiro sensei as what is called the Iwama style which maintains this feature of bukiwaza.
Other teachers derive their sword training from different sources which explains the variation in approaches. The litigious societies some of us live in are also a contributing factor.
So now the discussion is getting a bit more open I still don't see much of 'pro's and cons' being mentioned. I see references only to who does and who doesn't and that's about all.
Plenty of why to use a tsuba and the theme is basically once you've been hit or could have been.
This is where I part from the standard 'logic' for to me that is fear based. Now, I'm not saying it is wrong to use a tsuba but when the reason is 'because my hand would have been smashed' then I see no technical reason given, no technical reason why the hand got hit, as if there is no way out of such a position. Held as a solid belief I thus disagree, however that once again doesn't mean it's wrong to use a tsuba. It all depends on what you are practicing and why.
With a live blade then it would be a bit of added security rather than a necessity, in other words I would tell anyone don't rely
on it. You can learn whatever fancy ways of using it you like but if you ever rely
on it then you may get a nasty surprise somewhere along the line.