I don't think there are any "official" response times across the industry. Each agent/agency is going to be different, and yes, it varies by workload. Some agencies will give you expected timeframes, and it does seem to be accepted that three months is what you wait before nudging if the agent has pages.
And it doesn't always take that long. Last time I started querying in April, had positive responses by July, and was contracted by early December (although the verbal agreement happened sooner). So about 7 months from first query to representation.
Honestly, I don't think it's gatekeeping. Most agents are
thrilled to come across work they think they'll want to rep. But the act of selling books for people they're already contracted with takes a lot of time and energy. Queries just aren't at the top of the stack, and with the volume they're dealing with, no matter how proactive they are it's going to take time.
It's all much faster than what happens after you sell a book to a publisher, though.