No more pages of charts to flip through to know exactly what every check should be. If failing the task doesn't change the plot in any way, I can tell the players they succeed even if they need to climb a cliff with heavy armor.
I think this is a good way of handling action resolution. But I don't think it is unique to 5e. I run 4e this way (and from memory the 4e DMG2, and maybe also the 4e DMG, call it out as a GMing technique). And I imagine other editions could be run this way also. In his DMG, Gygax suggests fudging secret door detection rolls to make sure the PCs find their way into an interesting part of the dungeon. That is an early version of the same technique, way back in 1st ed AD&D.
1E had flavor. It was parsimonious...but it was there. The 2E core books where relatively dry and style free. So where 3E and 4E, which also took a smaller, flatter, less inspiring approach in general. (here is your little vale and your few little gods).
I don't know the 2nd ed AD&D books well enough. I agree that 3E is rather dry. I don't agree that this is the case for the 4e PHB, which has a history of each of the races (which taken together give you a history of the mortal world) plus other bits and pieces scattered through the class descriptions (especially clerics and warlocks).
In fact, the presence of that default flavour in 4e is one of the more frequent criticisms that I see.
Of course you can house rule in 4E, it is just more difficult than in some other editions because of how tightly designed the game is.
I haven't really seen this, but then I'm not sure what you mean by "house rules". Is using Twist of Space (7th level wizard teleport attack) to free an NPC trapped inside a mirror (something that happened in my game) a house rule or not? The game's tight design supports rather than hinders that sort of play.
Making up new weapons, magic items, monsters, spells, etc is no harder than its ever been.
Plenty of people who post on these boards have house ruled skill challenges, recovery times, lingering injuries (based on the disease/curse track), etc.
To be honest, 5e strikes me as being quite similar to 4e in many respects: fairly tightly balanced class abilities (including the spell designs, with a few apparent exceptions), a very general non-combat resolution mechanic based around broadly-defined skill and stat checks (5e has a handful of slightly more pedantic skills), and integrated encounter design and rest/recovery mechanics.
What sorts of house rules do you have in mind that you think are easier in 5e than 4e?