Doesn't the spell require animals to be able to communicate verbally? As far as I know, spiders are unable to talk to each other using sound.
You gain the ability to comprehend and verbally communicate with beasts for the duration. The knowledge and awareness of many beasts is limited by their intelligence, but at minimum, beasts can give you information about nearby locations and monsters, including whatever they can perceive or have perceived within the past day. You might be able to persuade a beast to perform a small favor for you, at the GM’s discretion.
With that order of words I would conclude that It does require you to be able to communicate verbally in general and the animal to be able to hear you. But it says nothing about the animal's ability to communicate verbally within its kind. Otherwise the spell would be quite useless, since all animals rely heavily on other ways of communication than verbal.
Plus there are other spiders in the game, with whom you can talk. So the matriarch looks more like an oversight than a design decision.