Have you checked it recently? The construction has made a mess of the walking routes for quite a while. I used to take very close to the route you show, but the curve to the northeast right after you cross World Way to get away from the terminals has been blocked by construction for most of the pandemic. I've drawn in red the route I've been taking - the vertical part follows the "this way to LAXit" route, but then crosses over toward the administration building (the circular driveway) after crossing center way and then follows the sidewalks there to the flagpole park. There's been a flag person at the center way crossing as close to 24/7 as I can tell (at least as late as 1 am). I've also put in blue what I think was the route to the LAXit lot last time I went that way - it's basically follow the red but continue following LAXit signs after crossing center way.

It's also possible to walk out east past T8 and onto the little island between the ramps and Sepulveda, then head north on the west side of Sepulveda, but unless you're going to the bus stop on Sepulveda that's probably the last choice of way.