Let's just take Steven Okert as an example.
One of your "low leverage" reliever examples. He's unimportant, because he only pitches when it doesn't matter. I used him because he's your first example listed.
Below are his game logs from April and May last season. A few highlights from Just two months:
5 blown saves (he entered with a lead and gave it up), all of which ended up as Twins losses.
3 holds ( 2 in 1 run wins and 1 in a 2 run win).
1 save (finished a 3-2 win.)
1 Win (in a 3-2 game, BTW.)
That's in just the 1st 2 months of last season. 10 usages that directly influenced winning or losing games.
Now tell us again Okert only pitched when it didn't matter.
https://www.baseball-reference.com/players/gl.fcgi?id=okertst01&t=p&year=2024
Scott Barlow you say?
April 2024: 7 holds, 2 Wins, 2 losses, a save and a blown save.
Just in one month..
https://www.baseball-reference.com/players/gl.fcgi?id=barlosc01&t=p&year=2024
I'm too lazy to do more, but I assure you, EVERY pitcher whonspent much time in a big league bullpen last year had multiple opportunities in games where their performance mattered.