Not with a typical amount and quality of potions
In my experience running and playing games, players have a fairly limited supply of potions. In Tier 2, each player typically has one or two buff potions and a small supply of healing potions.
In order to evaluate the change, we need to look at who you are casting haste on.
- Frontliners typically hit things and get hit in return. They benefit from all aspects of haste - an additional attack, +2 AC, and better speed and Dex saves. Their best use of the hasted action is probably to attack, but they may drink a powerful buff potion.
- Ranged DPS just does the hitting, ideally. They don't get as much benefit from the AC and speed, but the additional attack and Dex saves are still nice. Like frontliners, it's probably best for them to just attack.
- Rogues love being hasted. Using Ready, they are able to Sneak Attack twice per round (but not per turn). They are unlikely to do anything but attack with the hasted action.
- Backline spellcasters don't benefit too much from haste, even with the proposed change. They are now able to use their action to cast and still drink a potion, but most potions are better used on frontline characters. On turns where they don't want to drink a potion, their hasted action is typically wasted.
Optimal targets for haste also tend to be optimal targets for buffing and healing (the vast majority of potion effects). These optimal targets primarily attack on any given turn. Thus, with your proposed change, a hasted target deciding how to use their additional action will most often weigh "drink this potion" vs "attack this enemy once".
So how balanced is this decision? There are three basic possibilities:
- Attacking once is far better than drinking a potion. In this case, drinking a potion would be considered an underpowered use of the hasted action. Your change would affect the game very little.
- Attacking once is far worse than drinking a potion. Here, drinking a potion would be considered an overpowered use of the hasted action. Your change would affect the power of haste a lot, which may have other balance ramifications.
- Attack once is fairly comparable to drinking a potion. Each turn, a player is pretty likely to pick either one. If we take "attacking once with the hasted action" to be balanced, then this scenario would make drinking a potion balanced as well.
Note that I'm not focusing on the ease of potion use before and after this change. Why? Because it doesn't really matter; what matters is opportunity cost. The proposed change certainly makes potion use strictly easier, but does that make it underpowered, overpowered, or balanced? Given that haste already permits a single attack (which we assume is balanced), the determining factor is: how powerful are the potions?
Potion of healing
The classic healing potion; heals for 7 HP on average. By the time someone's casting haste, the party barbarian has 55+ HP and the fighter isn't far behind. Healing 7 HP isn't nothing, but it's unlikely to be worth forgoing an attack. Healing in 5e is largely a waste, unless the target is unconscious. (There are exceptions - big spells like heal and nasty enemies that kill when they drop you to 0 HP.)
More powerful healing potions exist, but they scale rather poorly. A potion of superior healing heals for 28 HP on average, the same damage a single player takes from failing a fireball save. The potion is also Rare, which places it (roughly) at the same power level as a potion of invulnerability. The latter has far more damage-removing potential when used properly.
Thus, I would consider drinking a healing potion an underpowered use of the hasted action.
A potion of invulnerability seems like an excellent magic item. Resistance to all damage for a minute! Unfortunately, the tradeoff is skipping (most of) your first turn, since its short duration doesn't allow for easy out-of-combat use. The outcome of combat is usually determined within 3 rounds*. Skipping a third of combat for damage resistance is a hard sell. Skipping one attack, though? That's more like it.
Even longer-lasting buff potions, like a potion of fire giant strength, are more attractive when used with haste. The players sometimes don't know when they'll encounter a big dangerous fight, so they never pre-drink potions. Given the cost of drinking them in combat, even longer-lasting buffs rarely get used.
Thus, I would consider drinking a powerful buff potion an overpowered use of the hasted action.
But wait! If a character drinks a buff potion, you also have to consider the opportunity cost of no longer having a buff potion. If potions are properly limited, characters can't simply drink them whenever they want. With those limitations in mind, I would consider drinking a powerful limited-use buff potion a balanced use of the hasted action.
Your proposed change will certainly increase the power of haste, but the best targets are usually better off hitting the enemy in the face, than downing a potion. Buff potions become much more usable - and thus more powerful - but as long as you don't give them out like candy, overall game balance should not be harmed.
* "The outcome of combat is usually determined within 3 rounds" is based on my personal experience playing in several hundred Adventurers League games (Tiers 1-4), and running Ghosts of Saltmarsh (AL) and Out of the Abyss (non-AL; extended into late Tier 4).
Combat may often continue past those 3 rounds, but the result is a foregone conclusion. Wizard casts fear on the first turn, and 3 of 4 enemies fail? Stick a fork in that encounter - it's done! This effect becomes more pronounced the higher the character level.
This answer pegs the number at 4-5, but that's the entire duration of the combat, including any cleanup at the end. The DMG implies that 3 is typical.