A lot of reasons, Flavor, Better initial spell list, a lot of abilities like Bardic Inspiration I just don't want, etc. Bladesingers have been my favorite since 2E ( see my Handle ). I haven't been happy with the Lore changes, but that's easily House Ruled. And, to be fair, the armor thing is easily House Ruled as well.
That doesn't look like a lot of reasons. ;-P
The first thing I would point out is that flavor is easier to change than rules, and bladesong is based in performance so that's not really off kilter as a bard.
The second thing I would point out is that I don't think Bardic Inspiration ruins the concept. A gish that allows other characters bonuses to attack rolls, saving throws, armor class, and damage suits the concept.
Spells is worth discussing because it depends on what a person wants out of the spells in the first place. Both classes share several spells and both classes have spells the other does not. Magical Secrets means bards can take any spell on the wizard list, but not all of them because of the different spell prep rules so wizards can still have many spells the bard would never have.
I'm curious what spells you think wizards need for your bladesinger. You can obviously house rule as you want at a table you run, but that's not really relevant to the default rules.
Song of Victory and Battle Magic are the same ability. Extra Attack is the same ability for either. Bards don't get that 10th level subclass ability, but Song of Defense prevents damage for spell slots where a bard can spend spell slots on healing spells from a much earlier level to heal damage instead of preventing it so the difference is minor.
The valor bard starts with light armor while the bladesinger starts with no armor and continues with no armor. The valor bard adds medium and a shield, which is something you're looking for. The valor bard also gains more weapon proficiencies and already has that additional skill the bladesinger picks up, but with more options. Both can use a weapon as a spellcasting focus.
The bladesinger can only use blade song three times per day, typically, at 3rd level. This can also be ended by any incapacitation effect. Bladesong grants the AC benefit, using INT for attacks, and using weapons as a spell focus so that's all lost with the loss of bladesong. This makes it a temporary gish application that the bard isn't losing out on. The only subclass advantage the bladesinger grants is INT bonus attacking when bladesong is active. Either class can pick up Shillelagh somehow.
So maybe bard isn't for you for some other reason. It's your character and I'm not going to tell you what you want. But for other people I would probably continue to advise them to use a valor bard as a gish option similar to the bladesinger wizard. The real difference comes from base class features.
