they would be used for inspection of track (another name for them is an inspection car) where speed is the most important. car is faster.
you could also carry a small amount of tools and materials easier than walking and carrying.
Steel wheels on steel rails have extremely low rolling resistance; your biggest loss will be wind drag, but with an appropriately geared crank mechanism, you should be able to travel faster than a walking pace while still expending only as much energy as walking.
As noted, you can also carry substantial amounts of cargo.
Conventional railroad grades (ones that rely on the steel load-bearing wheels for tractive force) are rarely very steep, so a handcar can be used almost anywhere.
Due to the low rolling friction of steel wheels on rails, a handcar needs a “parking brake” to keep it from rolling away when stopped on a grade. This may have been what people were seeing when they thought there was a lever to set the direction of travel. Thinking about this, some sort of ratchet mechanism might be helpful if you needed to start out going up a grade…but you could probably get the job done with a simple chock on a rope, and that would be cheap and fairly foolproof.
Somebody was getting paid by the minute, weren’t they?
I thought the link was going to be to this.
That may be what I saw. It looked like some kind of rack drive, but may only have been a stationary rack to engage the main gear to keep it from turning.
The one I saw really controlled the direction. Kick it one way, you went that way. Kick it the other, you went the other. The brake was a separate lever. My uncle, an old, retired railroad guy was with me, and showed me how it worked. He used to use that model. It was only about 3-4 years ago, so my memory of my uncle showing me how it worked is reasonably fresh. I didn’t look carefully at the mechanism, so I don’t know how it worked.
That said, I’ve only ever seen the one. Other models may well not have had direction levers.
At least no one has asked how you steer it!
I’m wondering if that is the brake mechanism. It’s not going anywhere on a level track, and you’d just need something to keep it in place on a hill. A ratchet type mechanism that allows it to move in only one direction would serve as a parking brake, and a directional control. So when you start up, you could only raise the pump lever or lower it depending on that setting.
I am a real railroad geek. That being said, if you’ve seen a physical specimen, in the company of a docent with experience on the specimen in question, and you have an adult memory of that visit that’s fairly recent than all I can say is that it’s fun being wrong because it means you’re learning something new. 
I know a lot about handcars, but there’s no way anybody can know everything. I’m curious about your geographic location, Cheshire Human; are you by chance on the right-hand side of the pond? I’m over here on the far left-hand side of the 'Mericas and most of my rail knowledge has to do with things Yankee. Do tell. I don’t know why designers or engineers felt the need for a reversing lever, but clearly they did at some point. Now I have a new research project!
Good on ya!
MM
Could it just be a directional brake as indicated in my post?
Yeah. I suppose that would do the trick, but CH seems to indicate that they were separate on the model she or he saw.
I’m in New York. My uncle worked for CN out in the BC, Alberta, Sask region. I just drove by that museum this summer, and there’s a new building where the track and handcar used to be, so I don’t know if they moved it, or no longer have it. They’re closed for the season, so I can’t go there and look until next summer.
ETA: Maybe it has the lever because it makes things easier in the mountains. My uncle worked in the Rockies. That’s just a guess.
Even I’m not THAT dense. 
Re. how to steer a rail vehicle
When I was very young I was in Telluride,CO where they have a specimen of a “Galloping Goose” Self propelled passenger rail-thingy decorating a lawn. It appears to be based on a road-going school bus, but may have been purpose built for all I know. These were used to provide passenger service on the tracks that were built mainly to transport ore from the mines. I clearly recall being astounded that the driver had foot pedals, a gear shift, and a hand operated parking brake, but no steering wheel.
Back on topic, a high-low gear selector makes sense, and it might be simpler (one less shaft?) if one of the gears reverses the action.
At least you had the excuse of being a kid. 
I just remembered something. There was a ‘neutral’ position on that lever, straight up. If you put it there, the ‘seesaw’ handles worked just like everyone in this thread said the ones that lack ‘direction control’ work. You push down a little on your side of the seesaw, and if you start going the wrong way, you pull up, instead, and you go the other way. And the lever was very conveniently placed for kicking if you were in a position to work the seesaw, and it had a big oval-shaped plate welded on top of the bar, to make a nice big kick-target. Which leg you kicked with depended on which end of the seesaw you were on. I don’t remember where the actual brake was, but I do remember that all of the controls, such as they are, were in positions that were very convenient for anyone working that seesaw.
Camp town ladies sing this song do da do da.
Here’s one where you can clearly see the friction brakes.
This is an interesting handcar called a velocipede. Here’s a view showing the outrigger third wheel.
This site has some interesting information. It mentions the sport of handcar racing.
Handcars may have taken many forms so that manufacturers could patent their own product. However, in the last link it describes how handcars took a beating. The basic units would have cost the least and must have dominated the market.
I’m resurrecting this thread in honor of my uncle Frank. He just died yesterday morning. He’s never going to be able to answer my questions about this.
Sorry fellow doper, I hope he found peace and died well lived, remember him. Peace be upon you and yours
Capt