This is completely untrue.
I’ve done extensive price comparisons based the cost of paper and ink. For different brands and models of printer, I’ve done comparisons of the number of pictures you have to print to break even, compared to the cost of taking it to a shop and having them print it for you. I included discount online printers as well.
These comparisons include the cost of paying for all consumables—usually ink, but some printer models have additional consumables which you have to include. These comparisons also factor in the difference between the included inks and the ink refills (some printers ship with ink cartridges that are less full than the refills).
It is not hard to come out ahead, doing your own prints. When I was doing this comparison, it was only economical to go to the store if you were sporadically printing the smallest picture sizes. For a medium picture size like 8"x10", you could be using high-quality paper and inks at home and still spend less money than bargain online print services.
As a rough estimate, printing yourself with high-quality inks and papers can come out to around $0.02 per square inch, or $1.60 for an 8"x10" photo. Right now, an 8x10 is $3.56 on Shutterfly, or $2.99 on Snapfish. That’s a pretty stark difference.
So you save about $1.40 per photo, and you can use this to calculate how long it takes to amortize the cost of the printer hardware. For a $250 printer, like a reasonable low-end Epson or Canon photo printer, that’s just 180 prints.
YMMV. I expect most people don’t print photos at all these days, and only put photos on their phone. I personally don’t see the point in printing out 3.5"x5" photos if I own a tablet or phone with a screen that large. So, my expectation is that people who print photos are making albums of family photos, or perhaps they’re amateur photographers or digital artists. Someone with a passing interest in photography or making photo albums, who prints, can easily blow through 180 prints to break even on a printer purchase. If you just have a few picture frames around the house you want to fill, obviously just go to a shop.
When I purchased my printer, I was comparing against more expensive local photo labs, rather than the cheapest online services. I was also comparing 11"x14" prints, which typically cost around $10 each, versus $3 at home. The purchase of a $700 printer is amortized after only 100 such prints, in this scenario.