It doesn't come up a whole lot in our games. Typically a potion is used to get a PC back up on his or her feet after they've gone down. Sometimes if there is no healing PC at the session and they don't have time to rest they might top themselves off with some potions.
Only in emergencies. I’m not sure we’ve ever bought one in 50+ sessions of our current campaign. That’s money we don’t want to spend. That leaves the ones we occasionally find along the way. We tend to save the few of those we come across until we know we are in or going to be in a dangerous spot and need more healing right then than we can get with spells. So if you are talking about long rests in an “Adventuring Day”, perhaps we’re in the ballpark of one per 15 to 20 long rests.
By the poll options I think your expectations aren’t aligned with the way the game is assumed to be played. Sure, if you have the cash and nothing else to spend it on, you could make potions a part of your daily health regime like rations and water, but that isn’t how I’ve ever seen the game played. If you’re coming from 4e (or a lot of computer games) that’s one of many things that is completely different.
I'm afraid none of the poll options work. We use them, but there's no way to break it down by encounter. A PC might have one or two potions, but we don't use them unless we need to. So we can go days without, or we could use them all during the actual encounter. It all depends.
We probably average one potion per long rest. Those really tend to come out when a character is downed and making death saves. Sort of like a magical AED defibrillator.
Adventurers League play tends to be different. Generally, long rests, and sometimes even short rests, aren’t a guaranteed thing. Neither is having a cleric with healing spells. Without dedicated healing or the ability to refresh those spell slots, I know I’ve had plenty of characters chug potion after potion in-between encounters.
I put one per short rest, but that's not very accurate.
My players are in Undermountain (old school, not DoMM) so they don't often find healing potions.
When they use healing potions, they tend to use several because its a big fight and they are in trouble etc. But, they only use healing potions maybe once every 10 fights, but then they tend to use all the ones they have.
As GM, you have control over how many healing potions they have access to. Yes, they are i the PHB, but if its a problem with them using too many, then you can simple say they are not available. It's very reasonable that the herbalist in town might only make 1 per week, and the party is not the only one buying them.
Our group likes to have them on hand, primarily to get someone back up (especially the healer) in combat. Outside of combat they're used if we have to push on without a rest and don't have any healing left other than potions. DMs in our games also heavily restrict how many we can buy in any given locations to keep us from just stocking up.
In the 5e game I'm playing in?
We guzzle healing potions like ail....Course that could be that before last session we didn't have a cleric, or really anyone with Cure Wounds. What we do have is GP. And some players use more potions than others.
In the 5e game I run? It's rare that the players use consumable magics.
It depends on the group and how many fights you have between rests. For example, do you have a cleric, druid or bard? Does anyone have the healer feat? A healers kit is cheaper than potions and heal more, although it is limited to once per short rest.
I wouldn't worry about it too much. I generally make a handful available at lower level and then let them purchase them as they are common items.