My thought making the bet 3x instead of 2x was because I did prefer to play heads-up versus multiple opponents, and I thought this was the better route to achieve that. It does kind of seem part of your reasoning may be more suited to a cash game though? I definitely agree you want to play more hands in position against weaker hands, and cash games should be treated together as part of one big session and not as distinct sessions, while each tournament is distinct. So when I think long-term, does that still hold given that tournaments are individual events and the goal is simply to maximize that individual tournament without regard to rolling over results? So, I sort of intuit this advice is better suited to cash games and not tournaments as much. Maybe there is no distinction on this? So I’m not disagreeing with you, just wondering if that point you’re making is equally valuable in a tournament setting as it is a cash game setting?
A golden rule in tournament poker:
The less chips you have, the more valuable they are to you.
And the more they need to be protected. It becomes so easy to get pot committed on small stacks, that any pot you enter has the potential to be your last. So you can't just splash around. Which is why small raises often do (and in theory should) get the job done.
And no to your cash game example. In cash games stacks are generally deeper, so you raise bigger pre to ensure the pots are bigger to get your value. Because all you care about in cash games theoretically is maximizing value. In tournaments you have to balance value with preserving your stack.
And blinds in a cash game are at best a recommendation. I've seen 1/3 games where the standard open is $20. Often the standard open and stack sizes are a better indicator of game size. Plus, people don't have to fear going broke in cash games.
But since people do have to fear going broke in a tournament, people should be playing a bit tighter especially around any pay jump. Long term holds true no matter what game style you are playing. Tournament poker is still poker, you just make adjustments to your variance tolerance since going broke is a problem.
As with all things in poker, everything depends. If you feel like the people you play against are going to call significantly more to a 2x raise than a 3x raise, then you certainly can take that into consideration. And you can always try sizes in-between. But when stacks are short, even a min raise can threaten a players entire stack.
As an example this guy calling your 3x raise with only 13bb is really quite bad. Because now on the flop if he misses and folds he just wasted 25% of his stack. Whereas if he just jammed pre, he either wins 4.5bb right then and there, or he gets to see all 5 cards and not get bluffed out and double up. And even just a 2x raise should be puttting him to that decision. The difference between doubling up from 13bb to 26bb and 10bb to 20bb is massive and not only for him, but for you. Because the shorter a person is when they double up through you, the more YOU still have left.
Just realize, that the more you risk the more often you will have to call shoves and you will lose more when you can't call a shove. Keep in mind that people don't want to go broke to a bigger stack when other players are shorter stacked or equal stacked to them. This means that people shouldn't be defending in positions outside of the BB very much.
I could go on and on. But I've rambled enough.