Yes, I think it's quite likely that tying and/or soldering* the spokes does nothing to the stiffness of the wheel, or has too small an effect to ever matter in the real world. I've seen no evidence to the contrary. I think it's even less likely that it adds strength or fatigue endurance.

So why do people do it? Or why did they historically do it? Which might be something different from why people do it now. Or the reasons given may have changed more than once over the history of the technique.

I see two main possibilities:

1. People have gotten this idea in their head, not based on evidence, that it increases stiffness (or strength or fatigue endurance). And no one noticed that it wasn't true.
2. Spokes break sometimes, and the spoke that's now only connected at one end can come out and flap around, hitting on the frame or the rider's leg, which might make you get off the bike and deal with it before you can keep riding. That can make you lose the race.

Of those, I think #2 is more likely, because it's provably true that t&s has that benefit. Plus it doesn't require all the people who did it to be ignorant of engineering, and too incurious about the world to notice that it doesn't work. OK wheel builders can have more than one reason, including product differentiation. But I think to do all that extra work, you have to believe it's actually doing something, or the technique would have died out. Oh wait, it mostly did...

I heard from one bike builder, very knowledgeable and experienced, and an actual engineer, who claims he's seen wheels where the spokes are deeply notched from rubbing against each other where they cross, which shows there's some relative motion there. That means tying & soldering would prevent that relative movement, therefore stiffening the wheel. I think he even said he'd seen the notching proceed so far that a spoke is sawed through and breaks, which tying and soldering could have prevented.

My problem with that is that in 60 years of working on bikes, I've never seen that. Doesn't prove it can't happen, but I think it implies that it's very rare. Too rare for me to worry about it, that's for sure.

Then there's the problem that there's no evidence that stiffening the wheel would even be a good thing. I can easily imagine that it could just as well make the wheel worse in some way, so don't say "it stiffens the wheel" as if that settles why it's good. But that's a whole 'nuther can o' worms, maybe let's not get into that particular religious war! (I'm obviously not a fan of the "stiffer is better" hypothesis.)

* Please note also that some of the proposed benefits of t&s require the soldering part. The OP's wheel has them tired but not soldered. That would not prevent the spokes from sawing against each other, and it seems to me it would reduce even further the possibility that it's stiffening the wheel in any meaningful way.

What tied but not soldered does do however, is restrain a broken spoke from flailing around. I think that adds more credence to my hypothesis that possibility #2 (of the two above) is the "real" reason people did it BITD. Why go to all the trouble of tying them, without also soldering, so that the tied spokes can still move relative to each other? It's got to be because you don't care if the spokes move relative to each other. You're just restraining broken spokes, that's all.

Or you saw it on your favorite pro's bike, and you thought it looked cool.