What would happen to someone without platelets? (science)

No platelets at all? Zero?

They would have been dead long before you make note of them. I live in Canada. The platelet count has units (normally) of (140 - 450) * 10^9 parts per liter of blood. It's sounds like a lot, but too few has side effects. If you have (say) below 140 and about 100, you have Thrombocytopenia, which, if there is no organic cause, is harmless.

Below that and they worry. Below that and you have a rare form of pre Leukemia.

If it drops below 10, they don't give you any time to worry or do anything.  If you don't have religious beliefs against it, you are booked for a transfusion and they start looking at your marrow. Severe lack of platelets usually isn't your only symptom. Other blood properties are usually involved.

The problem is that leukemia robs you of all energy as well as not being able to have your blood clot. Blood clotting is a very important properly of blood. If your blood doesn't clot, when blood is exposed in places it should not be, it flows freely without interruption. You can bleed to death.

Below 10  and you really are in trouble. Death can't be far away and it won't be pleasant.