This can be done from within the Steam client

There is an option within Steam that toggles whether or not Steam starts with your computer. To change it, open up the main Steam window and select 'Preferences' from the 'Steam' menu, followed by selecting the 'Interface' tab, which brings up the following screen;

Simply uncheck 'Run Steam when my computer starts' to prevent Steam from starting with your computer. (Obviously if you wanted to Steam to start with your PC, you'd check this box instead of unchecking it!)

macOS

If, on macOS, this doesn't work for you, check for Steam in "System Preferences" > "Users & Groups" > [Your user] > "Login Items", and remove it via the '-' sign.

It's definitely by design. From this topic:

Tito Shivan (Steam moderator):
As mentioned, the client does save the age. And whether it's saved or not, they won't disable the step of having to click and accept before entering the game page. It's a feature to remove any responsibility from Valve in case anyone not meant to visits the page (hence lying about their age)
It's a way to avoid stupid lawsuits.

The thread also mentions law texts excerpts that may help to understand. For example:

2.2 If the audio-visual content contains material from a Mature-rated or Adults Only-rated game (or one that is anticipated to receive either rating), display an ESRB-compliant age-gate prior to access or purchase.

You can now skip that validation step entirely: