How can I automate at (close to) sample rate?

I’m trying to automate some filter sweeps, but the automation doesn’t happen at anywhere near the resolution I want.
I’m using 256 frame buffers, running Cubase 11 Pro on Windows 10 Pro using a Steinberg UR-22 interface and the Yamaha ASIO drivers.
Here’s my automation track, running at 144 bpm:

What I hear is just a couple of hits, and the sixteenths hits at the end all mush together like an open filter.
VST has been advertised as supporting sample accurate automation for many years, so how can I actually achieve this?
(Note that the sound I hear sounds like it automates even slower than the 5.8 milliseconds I would get from “buffer rate” automation.)

If I understand the situation correctly I would assume that you should either change to a higher quantize like 1/64 or just turn off snap to freestyle it. If you press “j” and try to draw in automation you will notice the difference

Have you tried switching the internal transition time between soft and quick?

https://www.steinberg.net/forums/viewtopic.php?t=135392

Or use LFO Tool.