Whoa, easy on this one - this one's a double/triple-edged sword!

It may lead to serious issues if not balanced with other principles, namely on time management, priority setting and task scheduling.

What are the dangers of tackling every task that comes along at once? Your focus for longer tasks is disrupted and important larger tasks are continually pushed to a later day.
So there's a strong counter-principle to the "2-minute rule": Keep your priorities straight. This could be as simple as old-school Eisenhower matrix or something more sophisticated Kanban board.
Some service providers - from your local watch shop to dentists, vaccine centers and fast food shops - actually have lanes or time slots specifically for walk-ins.
This allows for "batching" of small, transactional, low-focus tasks which is another time-tested organization principle for service workers. So block an hour every day for e-mail inquiries or, in this case, changing batteries.

All on the condition that you have enough work overall. If you're bored, you can always tackle according to the FIFO or FCFS principle: First In First Out / First come, first served.

I realize I'm not giving a good alternative to the stated principle, and my argument apply mostly for the provider, not the customer. And for customers, getting something done is fantastic. But you may end up favoring certain customers (small, somewhat urgent tasks) over others (longer tasks but also time-constrained).
I think it also neatly ties in with the "Let me know why others come before me" principle, so if a customer needs to wait for a small tasks to be completed, tell him it's going to take 25 min because there's prior tasks to be finished.

What about waiting at the counter? Some just want to check in, others want to order coffee and cake, some still need to register and other to declare open issues. Any best practices? 

I need to change the battery of my watch. So I go to a tiny watch repair shop. The person at the counter tells me it’s something quick to do. She says: “Do you want to wait here for five minutes, or do you prefer to come back later?”

So I wait five minutes (1) and get a new battery. I’m impressed last time I had to change the battery I had to come back the next day.

That reminds me of the two-minute rule by David Allen (3), and we can adapt it to the world of Service Design like this:

If you can solve a customer or user issue in the next five minutes, do it with the person right away. (3)

This staff member understands well that it takes more time to fill out a form with my personal information and then do the repair than do it right away. So the worker and I saved precious time and even some precious mental energy (4).

So let me ask you:

Are there tasks that you often delay in your service? Which tasks could you do right away to save time for the user and your staff?

(1) The staff member made the waiting time fly by thinking aloud. For example, she said: “Let’s open that thing... Hmmm... Okay, the old battery is healthy... Let’s recycle it...” You get the idea. It makes the waiting time more interesting, and therefore it feels shorter.

(2) This rule says that if you receive a new task within two minutes, it’s better to do it right away instead of adding it to your do list system. The time you’ll need to add it to your system and the time you’ll need later to get back into the context takes more than two minutes. — David Allen (2001). Getting Things Done: The Art of Stress-Free Productivity. Penguin Books. Available at https://extra.swissinnovation.academy/6xTS accessed 19 October 2022.

(3) Psychology researchers and productivity experts seem to agree that switching contexts costs a lot of mental energy. — Ian Hayne (2021). The Cost of Multitasking: How Much Productivity Is Lost Through Task Switching?. Wrike. Available at https://extra.swissinnovation.academy/zEIq accessed 19 October 2022.

(4) You can combine this idea well with other principles like "Let me know why others come before me" when you run a more complex service.