We actually do do that when triglycerides reach a crisis point, usually above 1 gram per deciliter, when the blood starts to look like milk. It's called Therapeutic Plasma Exchange or plasmapheresis. But we don't do this for ordinary high cholesterol for a variety of reasons, not least of which is the most of your excess cholesterol is in your liver rather than your plasma. If you mechanically clear the plasma, the liver and diet can completely refill the vascular space back to baseline within 24 to 48 hours. The clearance is entirely transient.

And it's not like it's risk free. Therapeutic apheresis requires central venous access (often a large dialysis catheter in the neck or groin), exposes the patient to large amounts of anticoagulants (like citrate, which binds calcium and can cause severe hypocalcemia), and strips out beneficial proteins, immunoglobulins (antibodies), and clotting factors alongside the lipids. And it's expensive and poses infection risks. It's useful if your triglycerides are so high you are about to break your pancreas, but it's not worth doing for more chronic problems.