I think I see a possible reason for your evaluation.
Could it be that you would be satisfied by a symbolic change? That you're irked because the class didn't even get "that" much offense.
I think you are equating my advice not to expect the Bladesinger to be effective in melee with me not being satisfied. I'm not upset or anything, nor do I think the Bladesinger is a weak choice. I don't recommend making a Wizard with a sword and throwing Bladesinger on and thinking you won't be ineffective though. A symbolic change wouldn't change that.
Because I have now made quite a few posts where I alternate between asking you to explain what is missing and discussing how there are fundamental problems with the urge to have the Bladesinger enter melee.
Yes, I notice you keep suggesting I'm not answering your question, but I honestly don't see what's missing from my answers. I will try again.
What is missing from the Bladesinger is a balance between offensive and defensive abilities. An effective melee class has a decent offensive and defensive capability. This does not change at all if the melee class happens to also be a primary spellcaster. Look at the abilities granted to a War or Tempest Cleric, there is a mix of offensive and defensive abilities. The addition of full spellcasting doesn't change that balance, and as such, both options work quite well as melee characters. The Bladesinger gets a whole bunch of abilities at level 2, and there are multiple defensive abilities and no offensive ones.
If the Wizard had a bunch of good melee offensive boosting spells, they could easily take the place of any subclass specific abilities, but unfortunately, the Wizard list isn't really intended to do that (unsurprisingly), unlike spell lists like the Paladin or Cleric for example.
When the Bladesinger actually gets an offensive ability (first one at 6th level), it is incompatible with the Cantrips that were introduced to multiple classes (including the Wizard) to enhance melee capability. The first offensive ability that the Bladesinger gets that actually works with the cantrips is gained at level 14, after adding even more defensive abilities to the class option.
So to frame this specifically to your inquiry:
1) What is missing: A balance between offensive and defensive abilities
2) How there are fundamental problems with the urge to have the Bladesinger enter melee: There is no problem (fundamental or otherwise) with the urge to have your Bladesinger enter melee. That's what a Bladesinger is supposed to do thematically. The problem isn't with the urge, it would be with the expectation that you will be reasonably effective doing so. That expectation will not be met unless you play with the build to make it work.
I'm OK doing that. If I want to play a Bladesinger who enters melee, don't worry about me, I can make it work and be effective. However, anyone who just gives their wizard a melee weapon and takes the Bladesinger option may find themselves with a character that doesn't do what they were hoping.
What I mean is that the only way I manage to understand your position is if you're not really looking for the kind of offense that would make a minmaxer's Bladesinger enter melee; you're only looking for the kind of offense "bone" that would remove any lingering sense of being ridiculous for thinking to enter melee.
If you're having trouble understanding my position, let's avoid speaking in metaphors. I think characters who are made as "melee" characters should have comparable offense and defensive abilities. They don't need to be equal, but if one character class is reliably doing significantly more damage consistently than another, to the point where the damage the other is inflicting becomes largely irrelevant, that's a problem, because (at least in my experience), part of the fun of roleplaying is the feeling of being part of a team and contributing to it's successes.
Let me give you another example. The Blade Pact Warlock is absolutely fine if you take a level of fighter as your first level. Once you do that it performs well and a player will likely be satisfied with the results. However, why is a trick (the one level fighter dip) necessary? Because the base class fails to achieve what it should in regards to a Blade Pact Warlock. A Fighter 1/Warlock 9 should be no better than a Warlock 10, but if you are going the Blade Pact Warlock route, it is, and significantly so.
The issue with the Bladesinger is similar. Like the Blade Pact Warlock it's absolutely fine if you take the appropriate multiclass. The problem is that many players won't know to do that, and it could lead to disappointment.
perhaps you're not actually planning to build your Bladesinger around melee offense.
If I do play a Bladesinger, just like any other melee character, I will be looking for a character that is passable in offensive and defensive capabilities. I don't really concern myself with optimizing one over the other, I want them both to be good enough to be effective.
Perhaps you only want to look good when circumstances force you to take the melee attack action?
If I make a character that is going to use a melee weapon as its primary attack form, yes, that's when I want to look good, or at least look OK.
Just trying to express my frustration:
I still honestly can't understand why you are frustrated. I've attempted to answer your questions as honestly as I can.
I can't get a grip of what state that would make you enter melee that would still be balanced
I would enter melee with any character that is supposed to be a melee character that is decent at it. By "decent" I mean comparable with other melee characters. If you are doing less than 1/2 the damage, you aren't comparable. 70% as good? 80% as good? These are reasonable expectations IMO.
so one option is that none exist.
If you go over my past replies as well as this one, I've given many examples of things that would work. So where is this coming from?
I can't seem to get you to discuss my stance that any offense that's balanced for a full spellcaster wouldn't be used.
We can discuss it, but your stance is wrong. Full casters go into melee all the time. Clerics and Druids are both full casters, and they are often mixing it up face to face. The idea of playing a spellcaster that goes into melee combat is you want to prepare spells that enhance your melee capabilities. This is what a Paladin who casts Branding Smite is doing. This is what a Cleric who casts Spirit guardians is doing. This is what a Blade Pact Warlock who casts Hex is doing.
This isn't 3rd edition anymore when spellcasting is objectively better than swinging a weapon. Spellcasting has been tempered. The idea of playing a spellcaster who goes into melee is to have a melee character with some tricks up their sleeve. It's a concept, and maybe it's not the most powerful concept for that class, but as long as it's still reasonably effective, a lot of people will want to play it because they like that concept. My concern is that a Bladesinger without preplanning isn't going to be reasonably effective in melee.
Of course, it would be perfectly balanced to add offense to a class where it is suboptimal to use said offensive in the first place.
There's nothing wrong with suboptimal. There's something wrong with ineffective. These are not the same things.
Look at all the Blade Pact Warlocks out there. I've seen several played, and lots of testimonials of people who've played and enjoyed them (after taking 1 level of fighter). It is a perfectly reasonable optimization point to show that the same character could do more damage if it concentrated on Eldritch Blast instead, yet there are all kinds of Blade Pact Warlocks out there swinging weapons in gameplay, why do you think that is?
I guess I just don't see the reason for bringing up the grade if that were to happen, since the slightly-better offense would still not be used?
I can't speak for you, and maybe you will never use a character with a weapon if you have spells at your disposal. Speaking for myself, if I play a Bladesinger, I will be swinging a weapon (probably two, but the point stands). I will play with the build to make it effective, and likely dip fighter or paladin to give it that "slightly better offense", but a weapon I will surely use.
I'm hoping that answers your questions. I keep thinking I've answered them to get responses that you aren't satisfied that I've answered them. If the above is also insufficient, maybe we should just chalk it up to inability to communicate.