Coincidentally, I have spent about a week trying to figure out this question, here is what I have come up with so far:
Firstly, we have to start with the basics. Because I'm not going to start from the Capstone and work my way down, that doesn't flow, and most people get fed up with the class well before the Capstone is an option.
HD: D6
Armor: None
Weapons: Daggers, Darts, Slings, Quaterstaffs, Light Crossbows.
Tools: None
Saving Throws: INT, WIS
Skills: Pick two; Arcana, History, Insight, Investigation, Medicine, Religion.
Primary Ability Score: INT
This means that Wizards are the squishiest members of their team, sharing the smallest HD, the d6, with Sorcerers and being restricted from using even the least of armors. Wis is a good save to prevent some nasty conditions, but INT is rare unless you are in a campaign with Mind Flayers. But against traditional damaging spells you are quite vulnerable.
Wizards also have a small and the least useful selection of skills out of their contemporaries (Bard, Cleric, Druid, Sorcerer, and even Warlock). Insight is their only Social skill, leaving them unable to convince, bluff, or cajole, without resorting to brute force or differing to a different character. (Even Fighters get Intimidate!) Medicine is practically useless, you only need a DC 10 check to stabilize (which can be compensated for with a healers kit) and Wizards don't inherently have any way to cure sicknesses even when they know what is making their patient sick. For the exploration pillar of the game they get Investigation, which is often forgotten in favor of Perception. Rounding out their skill list they have Arcana, History, and Religion. While Arcana and Religion can be used to identify monsters in the middle of combat, their main use is for your DM to provide a Lore Dump. Arcana can also be used to identify a spell as it's being cast, but you would never actually want to do that because it uses your reaction, which means you can't Counterspell the recently identified Fireball that is coming for your face. Additionally, the Primary ability for the Wizard is INT, which doesn't lend itself to many active skill options. The silver lining as a Single Ability Dependent Class (SAD) is you could pick up a decent smattering of a more useful ability. Though you will probably want to focus on CON as your secondary ability, seeing as how that keeps you alive and lets you Concentrate on keeping your spells running.
At this point, (after looking at that anemic weapon list if you really have to) you should probably realize that Wizards are for basically one thing, Casting Spells. Which is good for them, because casting spells, and casting them more often (and ASIs, I suppose, but really, everyone gets those) is the bulk of what you get for being a Wizard. Heck, you don't even see half of your Core Class abilities until the final Tier of the game, so you can basically forget about them.
To be quite frank, during the first half of the game, you are going to look at your spellcasting piers with envy, as they also get spellcasting, good to amazing proficiencies, and class abilities that are way more robust and flavorful than just casting a few more spells per day. Adding insult to injury, a Wizard isn't even necessarily the best at casting any given spell, thanks to said class features.
So what is a low level Wizard good for then? Basically, Rituals. Your soul low level advantage (barring subclass or race features) is your Spellbook, which lets you cast every ritual you know whenever you want without having to prepare it before hand. While other classes can also perform rituals, Wizards inherently have the most rituals, and you also start at level 1 with 6 spells of your choice. Allowing you to get a massive leg up on your only real competition for this role, the Tome Pact Warlock (Make no mistake, they will beat you in this department if given enough time)
Now, I'm not going to sugar coat this, most of the rituals are highly circumstantial, if not pointless. There are 19 Rituals that a Wizard can learn (out of 34 in the game). 8 First level, 4 Second level, 4 Third level, 0 Fourth level (seriously), 2 Fifth level, and 2 Sixth level. Seventh level and above spells are too powerful (in theory anyway), so there aren't any rituals of those levels. Broken down by School: Abjuration, 1. Conjuration, 3. Divination, 5. Enchantment, 0. Evocation, 1. Illusion, 3. Necromancy, 2. Transmutation, 2. And out of all those Rituals, about 9 are worth going out of your way for.
It doesn't sound all that impressive, but still having 9+ mediocre to good abilities on-tap whenever you are out of combat will more than get you by in the early game. And if you really need a solution to a problem, there are always spell slots that can be burned to get around them. The only issue is, of course, surviving long enough to get all the spells that make the Wizard into a force comparable to what everyone says they are. And it's not like Wizards are particularly bad in combat either, they just have to focus on defensive tactics, which some find boring, but to each their own.
Ultimately though, I think that people are just overhyping the Wizard due to prior edition biases. The only spellcasters that are actually unhappy with themselves are Sorcerers, mostly due to their under filled spell list and lack of spell choices at the endgame.
War Wizard, btw, is in my opinion the worst Wizard subclass.
It's certainly redundant and boring.