I am finding that my ES350 restores the drive mode I leave it in if "Eco" or "Sport" but not "Custom." Is this true for others?

I'm in SPORT all the time but it NEVER stays there after shutdown.

I think "The Book" makes it clear that the "mode" is reset after the engine is turned off.
I would think that changing the color of the speedometer would be enough of a reminder for most drivers of the selected mode.

It has a custom mode?

That is very odd... leaving mine in Eco mode "stays" in Eco mode or leaving it in sport mode does the same. Just not for the Custom mode.

From the "4-5. Using the driving support systems" subchapter's "Driving Mode Select switch" section of the ES350 (and ES250) Owner's Manual:
The ES300h Owner's manual for model years prior to the 2023 model year's introduction of the ES300h F Sport Handling model type (the one with Adaptive Variable Suspension along with S+ and Custom drive modes) doesn't have the custom mode language included.
​​​​​​​

Drive Mode "Sport" increased the fun but decreases the MPG results.
EPA/CAFE testing requires the test to be run in whatever mode the test vehicle starts up in.
Manufacturers do Not want to post the MPG results from a Sport Mode test cycle.
Thus: no default Sport Mode or Custom Mode w/ Sport throttle settings available at start up.

The F-Sport models have a Custom mode as well as a Sport S+ mode.

That's only on F Sport models equipped with the Adaptive Variable Suspension system. (See the "Vehicles with Adaptive Variable Suspension system" subsection under the "Driving Mode Select switch" section of the "4-5. Using the driving support systems" subchapter of any version of the (US and Canadian) Owner's Manual.)

For 2019 through 2022 model years, that is F Sport models equipped with the F SPORT DYNAMIC HANDLING PACKAGE. For 2023 and later model years, that is F Sport Handling models (where Adaptive Variable Suspension is a standard feature).