In the original Beastmaster movie the Beastmaster Dar has two ferrets whose primary function is to steel things in the way you describe. So I would say "order your pet to steel something" is intended behaviour.
The issue is spiders are not usually considered as intelligent as ferrets (and the rules kind of reflect that, a ferret has an INT of 3, and a spider 2).
It's up to the DM to rule if the spider is not intelligent enough to follow those orders, or if it is a magically intelligent spider. Personally, in a pet solo mission, I would expect the player to role-play as the pet.
Yeah, that's fair, I do remember beastmaster and his little ferrets were smarter than your average critter. It's the wolf spider so it's a 3 Int, so it's dog intelligence. If I imagine being a magical smart pet that blurs the line between pet and familiar. She insists she has telepathic control over the pet. I can't see anything in the description like that though. "It follows orders to the best of its ability" is what it says, which can mean anything you want.
There is something else that annoys us though, as I've had another co-DM sit in a few times and he
aggressively disagrees with how the ranger is using the spider pet. Maybe it is a situation where you need to see it in play, I dunno.
It's hard to put a finger on exactly, but maybe it's how the spiderpet is controlled as an extension of the PC in every way, like a remote control familiar. But it's her "thing" and a unique ranger ability that does set her apart from other classes, so I can't harp it too much, and I think if it was one of the allowed listed pets it wouldn't be as much of a problem, like a panther or badger or ferret.
Another instance, someone was temporarily blinded with goop in their eyes, and the player wanted to order the spider to clean the goop away. This happens a lot, ordering the spider to do...weird things...and I cannot decide how to handle it. I don't think a ferret, dog, mule, or wolf would know what someone meant to "clean Beorn's eyes off" unless we are talking about exceptionally intelligent magical creatures, but I thought these were forest animals, albeit resurrected magically. So, cleaning the eyes is what the "character" would do, but tries to use the spider to do it instead as a remote control extension of herself.
I tried to get the player to swap out one of the listed animals but she refused, saying she was attached to the spider I'd originally allowed. I did have to remove the bonus 2d6 poison damage and paralysis because it was way more than what other pets could do. The spider climb and 40' speed is still quite powerful and the spider stays out of melee range until it can drop on someone's head and bite.
It killed many an enemy this way, and critting with the poison bite was doing 6d6 damage (on a failed save).
All that aside though, I do like the ranger has a stronger "this is my shtick" now, and the pet has full actions, but where do I draw the line between animal and familiar? The player is a newbie and doesn't have any concept of game balance.
I know that my old seasoned players would never have the pet do things like this, they would think it's silly.