Seems like there are a few different questions/issues involved here. Going to try to distinguish them...
Firstly, I almost exclusively play paper Magic. I am on a bit of a hiatus from Arena (briefly returned to the game in December/January, but moving house caused me to drop it again for now). I will be back, but it's not my focus. I own a huge collection of physical cards and have spent no money on Arena, so my style is very different between the two media. On Arena, I've mostly played the starting free decks. I do have an MTGO account, but pretty much gave up on it because I found I couldn't get over the clunky interface). My attitude across the three is definitely different. On Arena, I am playing bad free decks and mostly just goofing off while listening to podcasts or whatever. I expect to lose most of my games, but get annoyed when I am put up against decks with lots of rares. On MTGO, I was mostly just cussing out the clunky interface and the chess clock. When I play face-to-face, I approach different formats with different mindsets...
-A lot of my EDH gameplay, especially in the past year or so, has been with other people's decks. I go out of my way to make crazy stuff happen, even if it kills me.
-I've been playing some Canadian Highlander, and my attitude in that format is more competitive, but still highly experimental. I try to win, but I also prioritize learning, rather than gunning to win every game nonstop.
-Not really involved with Legacy much these days and mostly it's been playtesting anyway, but my attitude in Legacy is 100% competitive, doing whatever legitimate thing it takes to win.
-If it's some kind of casual Constructed, I try to read the room and I've often deliberately given people openings. While I don't know if it amounts to outright throwing games, I like to give newer players more of a chance than they might otherwise have, especially if I'm using my own decks. Like if I tutor for something, instead of just picking the card I think will win me the game, I'll pick something flashy that presents a dire threat, but with more opportunity for counterplay.
-I'm bad a Limited formats, but I go 100% competitive anyway and try to maximize my chances of winning at every turn. Because I have the draft skills of a brain-damaged neanderthal, I don't actually win. But I do everything I can to
try to win.
Secondly, I actually rather like infinite turns, heavy countermagic control, burning through people, etc. I like powerful decks that do powerful things. One of my favorite, most memorable decks I played against was in a "casual decks tournament" where my friend built what was, unbeknownst to me at the time, a version of Zvi Mowshowitz's TurboLand Oath deck. He'd establish control, flood the board with extra lands, maintain a grip of countermagic, and chip away with Treetop Village. If his opponent ever played a creature, Oath of Druids into Battlefield Scrounger would set up an infinite turns loop with Time Warp and Krosan Reclamation. It was glorious and I was a bit proud of myself for beating him in the finals with my discard deck. We saw some other cool, possibly broken stuff in that "tournament." I'm into that stuff. Wrote a bunch of articles about combo decks here at the CPA. It's what I like. But there's a caveat...
For good gameplay, decks should be balanced against each other. That's the reason I analyze formats and talk about formats so much. It's why I've been trying to help establish the groundwork for Tribal multiplayer formats here. Infinite turns are fine, but only if it's understood by everyone else that you can bring that sort of thing to the table, that they should expect to either be able to stop what you're doing or to do something else equally powerful. I think I've mentioned in the Magic Memories threads that one of my favorite decks in any competitive format was "TPS" in Vintage during the mid-00's. It was a deck with a fast combo kill and disruptive tools to protect itself, but its speed and disruption were balanced by the competition, and the matchups against the other top decks in the format were intense. It'd be silly to play Vintage TPS against, say, a Standard deck. Or pretty much any casual deck. In its environment, the deck was fun. Context matters.
I think I do understand you when you mention that you and your friends wouldn't play a Nexus of Fate deck because the group wouldn't like it. Well, consider this. The last time I played with extra turns cards was when I participated in a
Budget 2HG Commander tournament. I presented a brief summary in the thread about my performance in that event. Our opponents conceded when I landed a key Walk the Aeons (with the capacity to recur it multiple times under protection from my partner) while we were setting up for a kill with Approach of the Second Sun. But our opponents were themselves only a couple of turns away from setting up a kill using a Skullclamp engine. They'd already drawn tons of cards and shown that they could produce huge amounts of mana to keep a loop going, and if we hadn't started taking extra turns, they were either going to overpower us with massive (albeit not infinite) damage or win the game with Laboratory Maniac. In the context of that game, having already seen the guy across from me draw most of his deck with Skullclamp, I had no scruples at all about setting us up to take a bunch of extra turns in a row. None whatsoever. But if I'm in a casual game with friends and I know they're not prepared for that sort of thing? No thanks! No point in that. And while I know you weren't really around for them, Ferret, we had Tribal Game 14 and Tribal Game 15 here on the forums that kind of played a big part in teaching me to be more attentive to that. Because I
thought that my opponents were more prepared for the broken stuff I was trying to do, but they weren't. I learned that for casual games, I'd rather make err on the other side of the spectrum. I don't really find it frustrating to show up with a weaker deck and lose against something broken, not compared to being the guy who ran the table against opponents who weren't equipped to stop him.