Does a step counter work without internet?

Yes. The iPhone counts steps with a motion chip and stores the result in Apple Health, both on the device. An app that reads Health will show today's number with no signal at all. Only the parts around it - friends, leagues, advertising, backup - need a connection.

Does the iPhone count steps in airplane mode?

Yes, and it is the easiest way to prove the point: switch on flight mode, walk, then open Health. The steps are recorded. Flight mode turns off the radios, not the motion sensor.

Do step counter apps drain the battery?

Counting steps costs almost nothing, because the chip that does it is designed to run all day on a trickle. What costs battery is GPS and an app that keeps waking a radio to talk to a server. A step counter needs neither, so an app that asks for your location is worth a second look.

Does a step counter need GPS?

No. Steps are counted from motion, not position - you can count a thousand of them pacing one room, where a location fix would say you had gone nowhere. GPS belongs to route tracking, not to step counting.

does a step counter need the internet?

No — and you can prove it in a minute. Turn on flight mode, walk to the end of the street and back, then open Health. The steps are there. The counting is done by a chip in the phone and written to a database on the phone; there is nothing in that path for a signal to be missing from.

What actually needed the network, then

If an app goes blank on the Underground, the thing that stopped was never the counting. It was one of these:

What works with no signal, and what doesn't
part of the app	needs a signal?
counting your steps	No. Motion chip to Apple Health, both on the device.
showing today's number	No, for an app that reads Health. Yes, for one that keeps its count on a server and asks for it back.
the widget	No — unless it is drawing something fetched from elsewhere.
friends, leagues, challenges	Yes. Other people's numbers live on somebody's server by definition.
advertising	Yes. Ads are fetched. An app that is visibly unhappy offline is often telling you how it is funded.
backup to your other devices	Yes, and this is the one worth having — it is how your history survives a new phone.

That fifth row is the useful one. Without ads and without internet are the two most common things people ask a step counter to be, and they are frequently the same request wearing different clothes.

Which makes this the battery question as well

The chip that counts your steps is built to run continuously on almost nothing — that is its entire reason for existing, and it is why the counting is always on rather than something you start. GPS is a different order of magnitude. So is a radio that wakes up to talk to a server.

A step counter has no use for GPS. Steps come from motion, not from position: you can count a thousand of them pacing one room, where a location fix would report that you went nowhere. If an app asks for your location in order to count walking, that request is the answer to the battery question, asked before you have even installed it.

How to check what an app is doing to your battery

Settings → Battery. Scroll to the per-app list and switch it to the last 10 days. A step counter should be near the bottom, in the same territory as a clock.
Settings → Privacy & Security → Location Services. Find the app. Never is the right answer for something counting steps.
Watch the status bar. The location arrow appearing while you walk, with nothing map-shaped on screen, tells you where the power is going.

The one place offline genuinely costs you something

Your history has to live somewhere. Health data can travel between your own devices through iCloud, which needs a signal at some point — not to count, but to arrive on the next phone. An app that keeps its own separate copy on someone else's server is offering the same thing at a different price: their backup, their outage, their terms.

Worth knowing which of the two you are getting before the day you replace the phone.

Where we stand

Angel Step has no server to reach. It reads Apple Health on your phone, draws the number, and that is the whole of it — in flight mode, in a tunnel, in a country whose SIM you don't have. It never requests your location, so there is no GPS bill to pay, and there are no ads to fetch. What it keeps and where is written out on the privacy page.

It isn't on the App Store yet. If the neighbouring question is the one you actually came with, step counters without ads covers what a listing does and doesn't admit, and do you need a watch covers the other half of "what does this thing require of me".