i dont know what lv you are playing your campaign, but it seem weird for me that a barbarian with 2 attack, a d12 weapon, + 1 and 1/2 str, + probably 10 GWM x attack (without count that if he kill anything he have a 3rd attack, and not considering eventually brutal crit) + rage, get less damage output than a rogue with one attack and just +5 dex to add to sneak attack. Or he is very unlucky, or he is not playing a dpr barbarian, that if half orc or half elf, with the right feats, its one of the best dpr in the game.I have a player in one of the games I run that is convinced that his barbarian is excellent on damage. And he's pretty good. But he regularly gets out-done by the rogue. Which makes perfect sense - if one weapon class has better defenses than another, it probably doesn't also have better offense.
Please, get out of your head on the theoretical and listen to two people who have played bladesingers over an extended level range. Bladesingers are more tanky than striker when in melee, and that's okay. It's fulfilling an important role in the party. Trying to do things like kite with the mobile feat means that foes won't be wasting attacks on you and therefore you are doing less for your party. Intend to be there and get attacked.
The flip side of that is you're a full caster and your very tanky - you're not also the master of weapon damage. With your spells you will be acceptable, but trying to optimize your "decent" instead of trying to play to your strengths ends you up like that barbarian - upset when the rogue out-damages him.
About mobile you are probably right, indeed i writed also "(or maybe not)", but the idea in that build is to go to get the casters on the backlines before they can cast (dont need mobile feat for it that much anyway) while the paladin, barbarian, cleric and monk tank the others, its anyway an important role that with that speed bladesinger can do very well, having also proficiency in wis saving that is the most rolled saving agaist a caster and counting on things like counterspell, dispell magic etc etc he can totally make the enemy caster useless and dead in few rounds and in my opinion this is better than tank all the enemies fighters while the casters from the back keep casting or healing them. This said, i think shadow blade is good only on specifical lv of pure bladesinger (from 6 to 9 it is used but it also fight with haste and also blur + mirror image its a so strong defense, but for sure its the best weapon we have, than from 9 to 11 you probably use bigby's hand, at 11 if you have a good sword you can think to cast TT, at 13 its all up to you i suppose), while on other lv is better to cast something else with concentration and go with rapier (shortsowrd its even better in my opinion as its much easier to find magic shortswords than magic rapiers, and we need a good magic sword that add damage more than bonus to hit before lv 11) and BB, this till lv 13, where simulacrum can concentrate for you and you can easly cast SB or TT.