FWIW, I did some quick reading up tonight. I can't believe I've not paid attention to this in over thirty years as a DM. 5e states that a potion usually contains an ounce of liquid. A shot glass typically holds 1.25 to 1.5 ounces. There's 2 tablespoons per ounce, for another frame of reference.

Where did you read that? The PHB only says that a potion is contained in a vial, and the description of a vial says it holds 4 ounces.

EDIT: I see, the DMG, page 139. My math in the rest of this comment makes the assumption that potions must be about the same density as water, but this development can only mean one thing: potions are four times the density of water. This makes it more dense than a rock! This has no rules implications, but it's interesting!

Further, an empty vial has no weight listed, which means it must weigh less than a quarter of a pound (the smallest weight the rules account for). A potion weighs half a pound, which means that (assuming a similar density to water) there must be at least 3.8 ounces of liquid in the vial.

Granted, that's still only half a cup, but the rules say it takes an action to drink a potion. We need to flavor potions to account for that, not change the rules to fit our interpretation of how we want potions to work.

My workaround is to make it so that the PC can just cronch down on the vial with a bonus action. The broken glass deals a d4 damage, but the potion still heals the same amount it normally would.