Silver is listed officially as the rival.
And "gif" is officially pronounced like "jif" but you'll find me laying dead in the gutter before you see me saying it like that.
Let's Go Pikachu is officially a main series game, yet to a lot of people, they play differently enough that it's more of a side game.
Just because some people in some position of authority said it's so, it doesn't mean that it's a sensible decision. Especially in a situation like this where "rival" has many possible different meanings and "designated guy you beat a lot" (the one usage that would make Silver's designation sensible) isn't really an interesting one.
He is still technically the rival, even if he isn't doing a traditional gym challenge, he is clearly challenging gym leaders.
Is he
clearly challenging gym leaders? The one time he mentions gym leaders is at Olivine and that's said pretty much in passing. It's perfectly plausible that he just heard it from somewhere, or went there to try and steel Jasmine's Pokémon, or beat up the people in the lighthouse and discovered that Jasmine was there because of them.
There is no compelling evidence that he's challenging gym leaders.
But, regardless, the shared goal of taking down team rocket, and constantly evolving dialog between the two including their battle at victory road where he relinquished his anguish and challenges you to prove his strength as a trainer.
Nobody disagreed that he's strong. Silver is indeed one of the strongest trainers you meet across the franchise. At the very least, one of the strongest you meet in the games he's in.
He is definitely a rival. A rival is someone who is competing with another for the same objective or for superiority in the same field or activity.
He checks off those boxes.
OK, but I would disagree that Silver is competing with us for any given objective.
At the beginning of the game we see Silver get mad at us for letting the police know he was the thief, and then for interacting with him and subsequently beating him (Azalea and Burned Tower).
During the Rocket takeover he comes at us more from a place of obsession because he lost a lot, he believes our philosophy is bad and he just has a very angry personality than "I want to beat Team Rocket before you."
After that he has one more obsession encounter before finalizing his arc.
Silver is more of an angry stalker that would probably have absolutely zero qualms with you if you shared his "Only strong Pokémon matter" philosophy than someone that's strictly competing with you for anything, really.
I'll give it to you that he's supposed to be a foil to the character and I'll even say that he has a decent story arc for when he was made, but I don't think rival is a suitable word here.