Your declarations are nothing more than that. Declarations. You provided no objective proof that I made a bad choice. Your opinions and declarations are not objective truths.
Can you prove objectively, providing and proving your claims to be facts, that my choice is a bad one?
the banneret's features (aside from at level 7) hinge off of the fighter's base class features. you thus cannot use those banneret features if you use your class resources normally - you instead need to wait until you can trigger the riders to make use of those features. this concern exists for no other fighter subclass. not even the champion has to worry about not being able to use one of their subclass features because they thought now would be a good time to action surge.
rally cry gives out fairly anemic healing, not quite enough to negate a single regular attack from a creature of equal CR. that'd be fine if you could use it repeatedly, but you can only ever second wind once per short rest, meaning - like most 5e healing - the best time to use it is when you have 3 downed allies you can bring up AND you yourself are hurt. that's not exactly a common scenario. the samurai's temp HP and the battle master's rally are similarly anemic (and are temp HP), but fighting spirit also gives you advantage on all attacks for the turn and rally can be used multiple times per short rest, and neither require you to hoard your second wind.
the banneret gains expertise to persuasion, which is anywhere from a +3 (since it's gained at 7th level) to +6, depending on level. the samurai gains their wisdom modifier to persuasion (on top of the proficiency they likely took at level 3 or earlier), which will probably be a +2 or +3 if you care about using it, but they also gain proficiency in wisdom saves (or one of the other mental saves if they already had it) as part of the same feature. the battle master can gain a maneuver which lets them add a superiority die (on average +5 to +7 depending on level, up to 4-7 times a short rest depending on level and fighting style) not just to persuasion checks, but also to performance or intimidation checks. so while the banneret does gain the biggest and most consistent bonus to persuasion, the samurai gains a saving throw proficiency on top, and the battle master has more versatility with what they can apply their bonus to, making the banneret's expertise the most situational of these three features.
inspiring surge is just basically just commander's strike but the ally doesn't do extra damage and you need to spend an action surge. sure, commander's strike takes an attack and bonus action, but at least i don't need to time it with my action surge. it gets better at level 18 when two allies can attack, but that also makes it more situational.
bulwark might be good, and it's the only feature no other subclass can really match but it's only against mental saves and rides on indomitable. fail that save against dominate person? well, you could use indomitable, but then you won't be able to help the sorcerer against feeblemind next round. oh wait, the wizard just used fireball instead. oops.
and once you've used all your base class resources - second wind, action surge, and indomitable - what do you have? well, as a samurai or battle master, you have everything still. as a banneret...you have expertise to persuasion. that's cool, i guess, but at that point why are you even a banneret? why didn't you just take skill expert to pick up expertise to persuasion? hell, that replicates the ENTIRETY of royal envoy, except you can pick whatever skills you want.
but do you see what the banneret is? it's a discount bard. rally cry? (mass) healing word. royal envoy? expertise. inspiring surge? haste (from magical secrets). bulwark? bardic inspiration. you're trading your subclass to be a worse bard.
And whether a class is better or worse than another is further thrown out of whack because games vary in how much each pillar is represented.
how much a particular game represents each pillar is a criterion. you can evaluate whether an option is optimal or not by considering it.
It's impossible to prove that one class is objectively(always) better than another.
sure, but you can prove that one class is objectively better than another during the intended or expected gameplay experience (i.e. roughly equal pillar representation though perhaps leaning away from exploration, 2 short rests per long rest, all classes and subclasses allowed, feats and multiclassing allowed mainly because they're so common). that is the context most people evaluate classes in, because it's an easy reference point. obviously if you're playing a campaign where casters and feats aren't allowed and you're not going to do combat very often that expertise in persuasion might look nice. that's also a very unusual game.
Something else being overlook by the folks claiming one class is better or this choice is optimal and that one is suboptimal is that folks have different objectives. If my objective is to play a Fighter/Purple Dragon Knight, literally another choice I make other than Fighter/Purple Dragon Knight will be sub optimal in achieving my goal and therefore be the bad choice for me.
because that's a ridiculous objective when we're evaluating if something is actually good or not. we're not evaluating if the banneret is good at achieving the objective of playing a banneret. that's redundant.
it's also not what you stated your objective to be. stop changing your objectives.
Goals vary. Classes and choices can't be termed objectively(always) good or bad since folks don't get to tell others what their goals are.
we also assume certain goals in discussions unless other goals are stated. when people say "x subclass is bad", they mean in that assumed intended play state i mentioned earlier. saying "well actually that subclass is good because my goal is to play that subclass" isn't helpful at all.
it's also extremely narrow minded, because usually discussions about how x is bad are intended to discuss how x can be improved. by just going "well, actually, it's good because i want to play it", you're outright trying to prevent any discussion on how to make x better. you are, in effect, hurting your own play experience.