DOD Police attend the gates, and post 2009 all gates have active and passive measures to stop such run-away drivers- usually in the form of a channelized gate arrangement (passive) then an active barrier that any gate guard can activate from his station- usually a pneumatic gate some distance inside the installation that pop up on command. All designs are of course, location dependent. Stand off is key, and some bases/forts etc do not have much capability for appropriate standoff systems. Not having been to Campbell in a long time, I do not know what systems they are using, but it looks like someone/all of them failed to punch the button in a timely manner....
It’s been about 15 years since I’ve been to Ft. Campbell so I have no idea what setup they have at their gates. I’ve been to a couple of other bases more recently and they have the setup you described. Whether at every gate, I couldn’t say.
This has happened before at Ft. Campbell and I don’t recall getting follow-up reports afterwards then either, now that I think about it.