To your first question, no of course not. This isn't a binary system; there's plenty of room to avoid homogeneity unless we're visiting strawman town.
As far as minimums... let's look at what we have...
Barbarian - Pretty slim pickings, but... Totem warrior path is chock full of exploration pillary goodness - rituals, wolf totem, eagle totem.
Bard - I think we're good here, but for completeness... Expertise, jack of all trades, countercharm (since charm will more likely be in a social encounter), spells
Cleric - Heartily covered through spells, but also the Knowledge domain, and to a lesser extent nature and trickery domains
Druid - Again spells covers this, but we can add wild shape here too
Monk - Pretty sparse here too... All languages is nice, but at 13th level. Way of Shadow gets a little bit and of Elements to a lesser extent
Paladin - Spells
Ranger - Favored enemy, natural explorer, spells
Rogue - Expertise, reliable talent, light sprinkling in thief and assassin, or go whole hog with spells from arcane trickster
Sorcerer - Spells
Warlock - Spells, invocations
Wizard - Spells
Fighter ....Remarkable athlete (a half bonus to athletics) OR student of war (proficiency in one artisan tool) OR go eldritch knight for spells
I actually had trouble writing that last line without adding sad commentary. It's kind of embarrassing actually, how completely gypped they are unless they play wizard wanna-be. Monk is probably biting at the fighter's heels for last, and some barbarians, but the fighter wins this contest that no one want's to win.
As you can see it runs the gamut. I'd personally say barbarian is about as low as it should go, but YMMV. At least the barbarian feels like the 3 pillars were taken into account when building the class.
Exactly. The fighter, barbarian, and monk are all extremely lacking in utility outside of the combat pillar. It doesn't help that the fighter and barbarian's primary contribution to the exploration pillar is outdone by low level spells (everything you can do with Strength checks can be replicated with greater effeciency with low level spells like spider climb and jump). Hell, even your typical Strength check is rarely a timed issue, so can be repeated by a 10 Strength wizard until success is achieved, even for DC 20 checks.