And I never figured Ambassadors to be created in huge numbers to begin with. I like to think, based on the background, that these ships were the Galaxy-class of their time: the biggest and the best Starfleet could muster, for the big long exploration missions that made us proud. There wouldn't be more than a dozen like her in the fleet, like the Constitution or Galaxy (initially anyway) of their days.
While I suppose that's an ok in-universe explanation, it doesn't really fit with what we saw on screen.
Ent-C: Operating near the Klingon border in 2344.
Zhukov: Rendezvousing with the Ent-D to transfer a Vulcan ambassador.
Excalibur: Assigned to the tachyon detection grid.
Yamaguchi: Participated in the Wolf 359 battle.
None of these missions I would classify as "deep-space," nor would I infer that the class was made for deep-space exploration based solely on this.
Sure; but we're talking about a class of ship which is arguably several decades old by that point, and which would be slated to be supplanted by other explorers such as the Galaxy by the time we saw them. So, as they finished off their final five-year missions (or whatever), assuming Starfleet didn't want to send them out again but wanted to keep them around, they could get assigned the various milk run missions that support all the other stuff happening across the Federation. You could also argue that Excalibur, possibly being in spacedock at the time she was drafted, was between missions, which could also mean Yamaguchi could've been in the midst of refitting at Utopia Planitia when the Borg came knocking. Zhukov, well, may have had irreparable damage from when Barclay was in Engineering and shuttling Vulcan Ambassadors around was all she could muster anymore. :P
We could probably find an explanation for most of those, since it's a pretty small list to begin with. That is, except for those that end up as "troubleshooter" ships like the E-D that always end up wherever things need to be taken care of, which could be what the E-C was doing in sensitive space near the Klingon border in 2344, as that class was supposedly the Galaxy or Constitution of her time.
And while your point was that all the "other" Ambassadors with these four exceptions were all away on deep-space missions, simply because we never saw much of them, then how do we explain the absence of the numerous other classes we only saw a few times or not at all?
[Long and awesome list omitted]
That depends on what these ships are meant to do and how many were made as well. Looking at the US Navy registry, the current list of active ship classes with the most examples in that class are pretty small: we have Arleigh Burke-class destroyers (62 ships active), Ticonderoga-class cruisers (22), Perry-class frigates (17), LA-class subs (42), Ohio-class subs (18), and a whopping ten Nimitz-class carriers. There are a good ten-plus additional classes of warships active (~42 active per Wiki), but none with more than ten examples, and not even counting the number of freighters and logistical ships there are to support them. Altogether the six "top" classes account for about 80% of the active warships; the top three alone account for over 50%.
Extending this to Starfleet, there could similarly be five or so classes of ship which account for the vast majority of the fleet we see, while everyone else would literally be lost in the background. This is not a completist answer as we KNOW that no such ships were shown or rendered for the big fleet scenes, but at least there is a real-world context for why all those other ships were just not seen.