Eldritch Strike is unbalanced
In an nutshell, Eldritch Strike allows a properly built warlock to benefit simultaneously from the invocations that improve weapon attacks and invocations that improve Eldritch Blast.
Moreover, it lets the warlock harness more damage from Hex and Hexblade's Curse.
This all adds up to a significant boost in power.
Eldritch Strike comes with problems, some of which you mentioned in your balance assessment, but these problems are either minor, avoidable, or both:
- Eldritch Strike requires a free hand, so no shields.
- This problem is generally true for most melee warlocks anyway.
- Melee warlocks often get War Caster which basically solves this problem.
- Eldritch Strike requires the attack to hit, so it can't be used when no attacks hit.
- This invocation is mostly of interest for Pact of the Blade warlocks who get a second attack via Thirsting Blade, which greatly mitigates this problem.
- Eldritch Strike does not function with two handed weapons.
- This problem can't be worked around, but most of the warlock's damage comes from other bonuses, so the size of the weapon die is not a big deal.
- In most scenarios Eldritch Blast will be at a disadvantage if the warlock does not want to move back and trigger an opportunity attack.
- Enemies with a 10 foot (or more) reach do not cause this problem as the warlock has some leeway to move back without provoking opportunity attacks.
- The Crossbow Expert feat directly solves the problem.
- The Mobile feat solves this problem indirectly by letting the warlock move back without provoking opportunity attacks.
- Spells like Darkness or Shadow of Moil similarly solve this problem indirectly because unseen creatures do not provoke opportunity attacks.
- One handed melee weapons with the thrown or reach property, such as daggers or whips also mitigate this problem because the warlock can occasionally make an attack from 10 or more feet away. (Note: "attack with a melee weapon" is not the same as a "melee weapon attack")
- The Repelling Blast invocation also mitigates this problem at later levels, because if any of the beams hit, the enemy is pushed and the rest of the beams are no longer at disadvantage.
For a rough comparison, consider the Bonus actions granted by Two Weapon Fighting, Polearm Master, or Maddening Hex. Eldritch Blast becomes way stronger than any of those Bonus Actions once it gets multiple beams and Agonizing Blast.
Another rough comparison is Sorlocks as they can also cast Eldritch Blast as a bonus action. However, they must spend 2 sorcery points to do so, whereas Eldritch Strike is at will.
If I were to build a warlock to make the most out of Eldritch Strike I would create a Pact of the Blade Hexblade with Mobile, Agonizing Blast, Eldritch Strike, Lifedrinker, and Thirsting Blade.
In the endgame this warlock deals (1d8 + cha*2)*2 + (1d10 + cha)*4 = 71 at will damage per turn, but the warlock can deal another 6*6 from Hexblade's Curse and 6d6 from Hex for a total of 128 damage.
Roughly 1 in 5 turns the warlock can't cast Eldritch Blast due to missing both melee attacks, so the adjusted damage is 64 at will and 112 with resources.
Just for reference, a typical melee Pact of the Blade Hexblade with a glaive, Polarm Master, and the relevant invocations deals (1d10 + cha*2)*2 + (1d4 + cha*2) = 43.5 at will damage per turn or 72 with resources.
The warlock might occasionally get some extra reaction damage from the Polearm Master feat, but it won't make up the difference.