What can a familiar actually do?
As far as actions in combat are concerned, the quote you've included pretty much sums it up. A familiar can take any action that it is capable of other than the Attack action. This includes the Cast A Spell action, Dash, Disengage, Dodge, Help, Hide, Ready, Search, and Use an Object. Since familiars are monsters, they can also take any non-Attack action included in their statblock. For example, the octopus has an ink cloud ability.
Note that what exactly your familiar is capable of is up to your DM. For example, I would probably allow an intelligent lizard to scatter caltrops. An intelligent seahorse, on the other hand, I can't quite see administering potions. See also this 3.5 question for more information.
It's also worth noting that the familiar granted by the Warlock's Pact of the Chain can Attack, but it requires you to give up one of your own attacks.
However, I believe you are slightly missing the point of a familiar - just like in previous editions, its primary purpose is utility, not combat. For example, my players have used a raven familiar as a messenger and aerial scout. They have also mapped out an entire cave complex before setting a single foot in it using a lizard familiar. And finally, (I'm actually still slightly annoyed about this), they effectively skipped a covert, stealthy adventure through creative use of the lizard.
It is important to remember that a familiar is usually just a basic animal. Thus it is subject to the food chain. It is not an invincible extention of the player. Roving hawks, bobcats, other lizards, barracuda, etc can be used to curtail familiar movements if the DM wishes so.
The Sage Advice Compendium confirms that familiars can both use the Help action to grant advantage on an attack roll, as well as function as "allies" for the purposes of Sneak Attack:
Can the familiar you conjure with the find familiar spell use the Help action to grant you advantage on your attack roll? A familiar can’t attack, but it can take non-attack actions, including Help. As the text of the Help action indicates (PH, 192), the action doesn’t require you to be able to attack; you simply need to be able to provide some sort of distraction.
Does the familiar of find familiar count as an ally for the purposes of Sneak Attack? A familiar is an allied creature. Its proximity to a target can allow you to use the Sneak Attack feature or any other feature that requires the presence of an ally.
A familiar can be a very potent combat ally. If the familiar uses the Help action each round you can get advantage on your first attack roll each round against an enemy.
If you are a rogue, you also effectively get a permanent sneak attack since there is always and ally within 5' of an enemy.
And specific to Warlocks with Pact of the Chain, the familiar's low hit points are mitigated by invisibility (three of the available special familiars can go invisible).
In addition, for spell casters, you can deliver touch based attacks through your familiar while you are out of range.
This is not true; a familiar cannot help to give advantage on attack rolls. "A character can only provide help if the task is one that he or she could attempt alone." PHB 175
Find familiar PHB p240
Finally, when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell. Your familiar must be within 100 feet of you, and it must use its reaction to deliver the spell when you cast it. If the spell requires an attack roll, you use your attack modifier for the roll.
So while a normal familiar (i.e. not using the Warlock's Pact of the Chain feature) cannot use the attack option on it's turn, a spellcaster can use the familiar's reaction for that round to attempt to deliver a touch range spell cast on the caster's turn. The familiar attempts to deliver the spell using a touch attack using the caster's attack modifier. As it uses the familiar's reaction this touch attack is the only thing it can do (no movement, no anything) so it would have to already be within reach of the target on the caster's turn.