Yes, it is possible to be competitive as a melee wizard.
(Disclaimer: This is an answer with an example of how this can work. A melee wizard will be inferior to a pure wizard in most cases. The build has some cheese in it, which is to make it as viable and fun as possible)
It is possible for a melee Abjuration or Transmutation wizard to be competitive in melee as other melee classes. The problem is more akin to "What is competitive?" or "How to be competitive?" in terms of a class, especially in melee classes. The most simple way to be competitive is to play a role, a niche, that no other class can while being considered "melee". In the case of a Wizard the niche that is easier to fulfill is the "bag of tricks" or the "controller".
Another thing to consider is that being in melee is being in constant danger and, for a low HP class as the wizard is, it is a very dangerous place to be. Because of this particular reason, the wizard has to be very proactive with the spells and utilize them more strategically that a standard wizard.
The last thing to consider is: why melee? What do we want to achieve being in the front line as a melee wizard? We might want to be a tank or a DPS-ish character. Since wizards do not have the means to be effective using weapons and multiple attacks naturally, and under the schools traditions imposed, it will be hard to be a pure wizard; no multiple attacks, pitiful melee damage and no way to convert spell slots to weapon damage (aside from touch spells).
Being "pure" is a fool's errand, but there are ways to ensure 9th level spells and to be as effective as it can be in melee.
Therefore, to be a good melee wizard, what is needed is a strong base that helps us to fulfill the fluff of melee and the niche we want to fulfill. Since there are a lot of possible classes and race combinations, I will go with what I find the most stable from level 1. Obviously a melee wizard is not an easy class to play, so the challenge is great.
Race: Human, variant.
Attributes (+ Human adjustment) and point buy at level 1:
- STR 15 (14+1) or 16 (15+1)
- DEX 8
- CON 14
- WIS 9
- INT 15 (14+1) or 14 (13+1)
- CHA 13
Paladin 1, fighter 1, paladin 1. (Proficiency with CON save because of the stone)
Fighter 1, paladin 2. (Proficiency with CON save because Fighter)
Human feat: Dual Wielder (+1 AC and non-light weapons for two-weapon fighting)
Fighter 1 Fighting style: Two-weapon fighting.
Paladin 2 Fighting style: Defensive (+1 AC)
With this, the base is finished. It is a solid base for a melee wizard, and it fulfills several considerations:
- High AC: Plate armor + 2 AC from feat/fighting style is solid, almost as good as plate and a shield.
- Multiple attacks (2) with STR modifier in both. Wizards don't have a lot of spells that use the bonus action.
- Great survivability on low levels because of Lay on hand (Worth 10 HP) and
Second wind (1d10 + 1)
- DPS wise is not bad. With the help of Divine Smite (Paladin 2) we
can burn spell slots to increase the DPS as we see fit, plus
Multiple attacks, not that bad (It rocks at low levels).
- HP wise it is good at low levels to mid levels, and at higher levels
the wizard spells increase the defensive capabilities. Abjuration wizards have the upper hand in this regard.
Consideration to have: It will need the War Caster feat at level 7 (when your wizards levels start to kick in), thus making it impossible to have INT and STR at its maximum. They can be both at 18 (not bad) or one at 20 and the other at 16 with the hope to find an item to increases it at 19 or more.
From here the melee part is covered as well as possible for a wizard with access to 9th level spells. From here everything depends on the wizard to fulfill the desired niche.
From level 1 to 3: You are a melee class until this point. Nothing worth saying.
From level 4 to 6: You are a Paladin-ish melee hitter with access to the spell progression of a wizard.
From level 7 to 11: The wizard start to kick on, here is where most defensive spells start to shine. Illusions are great to create the ideal situation for your melee character and INT saves are rare. Summoned creatures are worth considering as defensive positions for you.
From level 12 to 19: So many spells, animate object is wonderful. Bigby's Hand is another treasure that cost's very little in terms of bonus actions. Mordenkainen's Sword is also very good as a controller since it is invulnerable. Also, low level spell slots are worth your time to increase your DPS as melee, just don't forget to leave some for shield and some defensive spells on dire fights.
Level 20: Wish: wish is the trump card at this level. Wish a Simulacrum of you and have tons of spells to back you up.
A melee wizard is very viable and competitive, not in terms of DPS, since at this point there are not many options. The niche to fulfill, as a melee class, is not easy fulfilled by any melee class.
A note for Cantrips VS Melee
Before level 17, most cantrips do less damage than the melee using the bonus action in most situations, but melee has the advantage that they can be strengthened with Divine Smite, especially on critical hits. After level 17, cantrips are better at doing consistent damage, unless you can get your hands in some nice magic weapons.
Consideration for the schools listed by the OP
Although my answer do not touch too much of the schools listed, it is easy (but not short) to find a much focused niche for those schools. The base build covers the melee part as good as possible, thus making any school viable, even more with the ones listed.