Pretty much. Just for a comparison there (granted it's not perfectly like for like) I booted up PS4 AC Odyssey, and PS5 AC Valhalla.

From hitting "continue" on the title screen to being able to control your character, Odyssey took over a minute, Valhalla was 13 seconds.

Saints Row 3 Remaster took 8 seconds from pressing continue to being in the open world.

I haven't really done any side by side comparisons with the two consoles, but have watched a bunch of Digital Foundry videos. They're effectively the same - Series X will sometimes have slightly higher resolution but slightly lower framerate. In most cases you'd have to go through the games with a tooth comb to really see any differences