I arrived at Civ IV after playing Axis & Allies, so perhaps that helped stacks be more intuitive to me. The overlap in stack mechanics between the two games has logic of its own and is also very rooted in odds. (Be mindful that the latter draws on WWII for its scenario and among its abstractions in no way engages the war crimes therein.)
One of the bonuses of large stacks is a clear differentiation between offensive stacks and defensive garrisons. In Axis & Allies, this means each power must maintain a garrison before launching an offensive stack or risk losing their capital or core territory: Soviets must defend against Axis land forces. Germany must anticipate an amphibious attack in Western/Southern Europe. As long as there is a German navy, Britain must garrison the British Isles. Japan must anticipate a possible American amphibious attack. Even America has to consider the possibility of Japan attacking the West Coast. Investing in offense and defense leads to producing different unit compositions, and fronts will aim to eliminate offensive stacks or collapse them into home garrisons.
As for Civ IV, it is the norm to see the AI garrison its cities and then build a stack to launch a war. This makes stacks quite legible, as stacks of a dozen or more units are almost always hostile to some party. In Civ VI, there is no analogous way to garrison units, almost any unit could be meant for defense or offense, and it is easy to deplete the AI's army only to find little resistance in their territory. This is comically present in the AI request to move forces, as it is not legible whether the player will invade them, liberate their territory, or transit to attack another civilization.
Stacks are also fairly indicative of economic investment. In Axis & Allies, normally all economic output goes into an army that will defeat the other side and survive as well as the naval means to transport and reinforce that army. Given that economic output is tied to the map and ultimately zero-sum, the loss of a stack or even a garrison can shift the economic scales of the war very much to one side's favor. Even trading forces is a bit of an art, as odds must be considered given losses on either side are decidedly economic.
In Civ IV, the loss of a stack can be very decisive, as launching a war necessitates a deviation from research or domestic investment. Deferring those priorities makes war a big risk: the loss of an offensive stack will doubly set the player back in available forces and also research/economy. In Civ VI, 1UPT generally precludes the economic shock of losing 30 units on a single turn. For better or worse, the economic angle is skewed by what fits on the screen. At the start of the game, this is quite acute: the army the AI may rush you with requires no investment on their part.
Another stack bonus presents as strategy in the form of dead zones. In Axis & Allies, a dead zone is where statistically there is no chance a force will survive an attack. In general, this acts as a front where a stack of doom threatens a home garrison. For example, as Germany advances on St. Petersburg, the Allies must decide whether they can survive by reinforcing or if the Soviets ought to retreat to Moscow. This dead zoning has many effects: for one, stacks tend to ward off adjacent territories, thereby securing economic output; for another, any stack must calculate whether attacking a smaller force will mean it finds itself in a dead zone. This tends to make stacks somewhat cautious. Whether you have five or fifty units, they would be better served as a garrison than falling in an enemy dead zone. Additionally, attacking below certain odds can have cascading effects on the board.
In Civ IV, the dead zone may not play the same role, but its strategic implications remain. Besieging a city only to lose 80% of the stack, leaves you open to a counterattack and does not bode well. A stack can also be tied up, such that guarding one's flanks falls to diplomacy as it is infeasible to have your stack on both offense and defense. With Civ VI's more tactical 1UPT, dead zones are normally the result of asymmetric technology rather than strategy. Given technological parity, there is no equivalent overwhelming of a hostile force. Instead, dead zones matter for units rather than armies.
Stack composition is also more varied than with 1UPT. In Axis & Allies, tanks, bombers, and to an extent artillery are stronger on offense, whereas infantry and fighters are stronger on defense. This variation necessitates some risky investment, with the Soviets as a good example. The Soviets must weigh how much offensive capacity they can invest in without diminishing their defense. More generally, an offensive stack can retreat into a garrison, but could be better served strafing the enemy. A garrison with just infantry and fighters can quickly lose control of core territories and face costly attempts to retake them.
In Civ IV, there are many unit counters, but suffice it to say that a stack with varied unit composition will be more robust than a stack with just one type of unit. This diminishes the ability for a stack of doom to besiege a well-composed garrison. While brute force sieges can and do happen, the trade-offs can be very large due to unit counters. This puts pressure on breakthrough units and tech advantages that make offensive wars less costly. In Civ VI, I am not sure army composition matters as much: ranged units are quite strong, anti-cavalry need to actually encounter cavalry, and many units provide asymmetric advantages. It seems more important to surround a city with units.
Overall, just sharing some thoughts on stacks and some of the trade-offs with 1UPT. Something general I have noticed is that personally I have a greater tolerance of reduced movement with stacks than with 1UPT. In Civ IV, it feels justified that a stack of doom should lumber through hostile territory. As for 1UPT, I definitely prefer Civ V to Civ VI in terms of increased mobility. For a strategic overlay like Axis & Allies or Civ IV, it makes sense that consolidation of forces comes with limited mobility. For the tactical map approach taken by 1UPT in Civ V and Civ VI, increased movement would seem necessary to allow for tactics to take place. A great modification to try if you have not already is double movement for Civ VI, which in particular aids AI cavalry. I would also gather that despite everyone's anecdotes of 1HP left, combat odds are more controlled in Civ VI than in Civ IV.
As for what happens to cities on capture, there is a pretty horrifying loss of population, but then again Civilization tends not to dwell on that. In Civ IV, you also lose all cultural buildings and city culture, which can be very frustrating in terms of rebuilding and reestablishing borders. In Civ V, there is loot and one can sell off buildings while razing. In Civ VI, city center buildings have to be repaired, and as with Civ IV razing is immediate. The idea that you can simply reassert governance seems a bit wishful... Realistically, cities changing hands in war have a very real chance of damage all the way up to being flattened.
If anything, I think Civ VI makes razing too easy. Given that it erases districts and improvements, I should think an invading force would have to put some effort into dismantling everything. It seems more a convenience so that orphan districts and improvements do not persist without a city. In all three games, I lament the trading of cities in war. Whether your own or another party's, it is sad to see beautiful cities reduced by an inability for one side or another to maintain control, though this is hardly ahistorical.