Why is giving the DM rerolls (or allowing the DM - who controls all the gods in the game, including the ones of luck and time - to intervene) on rare occasions a bigger rule change than giving the players some? [Not asking how folks feel about it, but why it's viewed as a bigger rule change.]
If the DM does it openly, then there is no difference. This is (one) part of why I emphasize secrecy so strongly: if it is done in the open, then realistically it is little different from giving players the ability to do it themselves.
If the DM does it secretly though... that's why it's a "bigger rules change." Because I as a player cannot possibly adapt to nor learn from such things. That's the whole point of making it secret! It's
specifically done so I should not, even in principle, be able to learn from it. With open rerolls or number changing (including open actions like "alright, this fight is just cleanup now, you've won, let's skip the boring bits") I can make decisions informed by this, e.g. if I know the DM tends to call fights when only a couple of scraggly minion-types remain, I can choose to focus my efforts on the bigger, badder enemies, or if I know that the DM doesn't call fights when there's some cost or danger to even a single opponent escaping, I can keep that in mind and try to avoid putting myself in a bad spot. Etc. If the DM fudges fights, though, I cannot learn any of this, and will make tactically unsound decisions because I literally don't know how the world "actually* works, I only know the filtered and glossed version the DM redacts behind the scenes.
Advice after hearing what people want sounds great.
If the only complaint about the system is that once in a while you hate strings of really bad/good luck, would parsimony suggest just putting a fix in for that? (Hence the furniture example).
Sure. Which is what I, at least, have been doing. I have said that "fudging" (secretly modifying rolls/stats of/applied to creatures already in play) does the thing people have asked for, but it comes with serious costs and flaws. Instead of doing that, I have suggested others use a mix of a small handful of other approaches, which collectively completely negate any need to "fudge" (as defined) while still achieving the intended goal. As a benefit, this mixed approach inherently (by design) avoids all of the serious costs and flaws associated with "fudging" (as defined).
To be specific, the mix I am speaking of is primarily:
Do things openly, be it explicit changes ("this fight is going way harder than I meant it to, sorry guys") or implicit ones ("calling" fights to skip the cleanup phase, telling players they don't need to roll they just succeed, etc.)
Do things diegetically, e.g. invoke divine intervention or a cursed/blessed bloodline or Luck/Fate/Death meddling etc. so there is a reason why things would be expected to change.
Do things before they have entered play, e.g. build safeguards into fights in advance, if absolutely necessary remove opponents or tweak encounter design, etc.
These are, I admit, a less "clean and neat" solution than just "you're the DM, rewrite (fictional) reality and invoke plausible deniability." But the gains are absolutely, I would argue
unequivocally, worth the price of (very slightly) higher complexity and effort.
My guess is folks who have heavily hacked one system will heavily hack most systems if you give them enough time.
I mean maybe? I spent literal years trying to kludge a Paladin into 3.X that didn't suck enormously. I dug up homebrew options and ACFs and houserules and custom spells and the official Prestige Paladin PrC and...just the list goes on and on and on, and I never came away satisfied. I then tried 4e's Paladin and fell in love, having wanted only and exactly one houserule for it ever (and that one only because WotC were jerks who took away a shiny awesome toy you ORIGINALLY had access to. Specifically, Call Celestial Steed for non-Cavalier Paladins.)
Sometimes people tinker because they just like tinkering, sure. But I find a lot of tinkering comes from
dissatisfaction with the thing as it exists, and it's not that rare for that dissatisfaction to be rooted deep enough that a person may not know the true cause. Again, I speak from personal experience. I spent years trying to make 3.X into the game I wanted it to be, and failed, in part because what I thought I wanted wasn't quite the same as what I actually wanted, and in part because 3.X's faults run so deep, you can't fix them without
gutting the system and effectively starting over (as Paizo's designers eventually admitted).