Unfortunately, your understanding is correct. I just tested this on Minecraft Java 1.18.1 and the turtle with the egg never lays it if unable to make its way back to the home beach.
I grabbed two turtles via boat and fenced them in on a new beach a short distance away. The wiki says "No matter how far away the turtle is, it always attempts to return to its home beach to lay its eggs after breeding", so I was hoping it would just lay it on the spot once the attempt fails. But it just waits indefinitely and only once I broke the fence did it go back to the home beach and lay eggs there.
Turtles must have hatched/spawned on your beach for it to be considered their home beach. The most reliable option would be getting Turtle eggs by using a Silk Touch tool. You could also try to just wait for some to spawn naturally under the following conditions:
Turtles spawn on the sand in the Overworld on beaches with daylight, but not in its snowy variant or stony shores, occasionally in small groups of up to 5 individuals.
In Java Edition, turtles spawn at light levels greater than 8, on sand blocks less than 4 blocks above sea level (typically at Y=62).
In Bedrock Edition, turtles spawn in groups of 2–6, at light level 7 or higher, at Y=60–67.
One guaranteed (though tedious) way to get Silk Touch is to just re-roll trades on a Librarian villager till Silk Touch is sold. It's not too hard to craft a lectern and they are sometimes already in natural villages. You'll just need to get enough emeralds, probably by trading with other villagers.