My impression is that Lockdown Mode is designed to guard against attack by highly sophisticated and targeted malware – at the possible expense of making your computer much less useful and convenient to use.
About Lockdown Mode – Apple Support
"Lockdown Mode is an optional, extreme protection that’s designed for the very few individuals who, because of who they are or what they do, might be personally targeted by some of the most sophisticated digital threats. Most people are never targeted by attacks of this nature.
When Lockdown Mode is enabled, your device won’t function like it typically does. To reduce the attack surface that potentially could be exploited by highly targeted mercenary spyware, certain apps, websites, and features are strictly limited for security and some experiences might not be available at all."
While cookies can be used to track you, Macs are not in the habit of treating cookies as executable code or scripts. Even if a site loaded a highly sophisticated piece of Mac-specific malware onto your system via a cookie, that code would be inert and harmless unless there was some other system compromise that let an attacker run it. (In which case the attacker wouldn't bother with the cookie and would use the other compromise to attack you directly.)
Note that even if you block all cookies, or purge them all on a regular basis, Web sites can still track you based upon things like your router's IP address (if you didn't have a return address, you couldn't communicate with remote sites at all), any accounts you are using on those sites – or data mining techniques that might not be 100% accurate, but which might be "good enough" from an advertiser's point of view.