The claim is wildly overblown
TL;DR While chromium and nickel (and iron) do leach into your food from stainless steel, they do so in forms and levels which are safe, even essential. Chromium and nickel are already naturally present in soil and food.
The author has likely confused this with extremely toxic forms of chromium and nickel.
There's no point to the abrasion claims if it isn't toxic, I'll just address the toxicity claims.
Yes, chromium and nickel are in your food...
There's no need to look to abrasion, most food already contains trace amounts of chromium and nickel. Cooking for long periods with an acidic food such as tomato sauce in a new stainless steel pan will leach more chromium and nickel. Repeated use reduces the leaching significantly until it effectively stops. The amounts are in micrograms (1 millionth of a gram).
Stainless Steel Leaches Nickel and Chromium into Foods During Cooking goes for the maximum leaching effect by cooking tomato sauce (one of the most acidic foods you're likely to cook) in brand new pans for hours. Even then the amounts were very low, and they drop with every use. They were unable to reach the Tolerable Upper Intake Level (UL), also known as Tolerable Daily Intake (TDI), used by the paper.
Note: UL/TDI is for chronic lifetime intake, it is not harmful to occasionally exceed the TDI. The paper demonstrates a worst case and unsustainable level of exposure. More on UL/TDI below.
...but safe forms of chromium and nickel.
Some authorities believe chromium and nickel are so toxic that none should be ingested;
Chromium and nickel are elements, but bare elements are rarely found in nature. They are almost always found combined together, often with other elements, into compounds. For example, the "lead" in your pencil is graphite, pure carbon. A diamond is also pure carbon. These are both "carbon", just arranged differently. The food we eat is also mostly carbon.
It's the same for chromium and nickel. There are many forms, some are extremely toxic, some are very safe. The author does not specify. Trace amounts of chromium and nickel are present naturally in the soil, water, and many foods, so I don't know what those "authorities" expect you to eat. I suspect the author mixed up the chromium and nickel present in stainless steel with highly toxic hexavalent chromium and nickel tetracarbonyl.
If someone is trying to make you scared of an element or "chemicals" or even "heavy metals", take it with a grain of sodium and chlorine; ask for details. Let's look at chromium and nickel in more detail.
Chromium
Chromium(III), trivalent chromium, is used by the body. A healthy individual should be getting about 20-30 micrograms a day. It's present in the soil and shows up in trace amounts in the food we eat. This is the form present in stainless-steel. Any scouring would produce chromium(III) oxide.
The chromium achieves corrosion resistance by forming a passivation layer of chromium(III) oxide (Cr2O3) when exposed to oxygen. The layer is too thin to be visible so the metal always appears shiny. It is, however, impervious to water and air, protecting the surface of
the metal beneath. Also, when the surface is scratched, this layer quickly reforms
From A Guide to Hexavalent Chromium Cr(VI) for Industry by the North Carolina Department of Labor
EFSA’s experts establish a tolerable daily intake (TDI) of 0.3 milligrams per kilogram of body weight per day for chromium III – which occurs naturally, is an essential nutrient and the main form of chromium present in food.
From the European Food Safety Authority (EFSA)
For a 70 kg person the TDI is 21 mg per day every day. That's 8 times higher than the UL used in the paper.
Chromium(VI), hexavalent chromium is extremely carcinogenic. However, this is of a concern for people who work with chromium, such as welders, or might be drinking contaminated drinking water. Hexavalent chromium is not present in stainless steel but it can be produced in its manufacture.
Hexavalent chromium can also be formed when performing “hot
work” such as welding on stainless steel, melting chromium metal or heating refractory bricks in kilns. In these situations,
the chromium is not originally in the hexavalent state but at sufficiently high temperatures undergoes oxidation (i.e., loses
electrons) to yield the hexavalent form.
Nickel
Nickel is also present in most foods, it's essential for plants, but it has not been identified as an essential nutrient for humans. Most that you eat is urinated out. While people can develop an allergy to nickel, it's not considered toxic at the levels leaching out of a stainless steel pan.
Nickel Tetracarbonyl is extremely toxic. It is made by reacting nickel with carbon monoxide (CO), if you have high concentrations of CO in your kitchen you have more pressing problems, and it decomposes very quickly in air. It also isn't present in stainless steel.
Experts increased the safe level, known as the tolerable daily intake (TDI), from 2.8 micrograms per kilogram of body weight to 13 micrograms per kilogram of body weight.
From EFSA
For a 70 kg person that's about 1 mg of nickel per day, every day. This is the same as the UL used in the leaching paper.
What about abrasion?
Knowing all that, how much steel would you have to scrape off your pan every day to exceed the TDI of chromium and nickel? Is it even plausible? Let's do some quick back of the envelope calculations.
The austenitic stainless steel used in most stainless steel pans is about 18% chromium and 9% nickel.
To reach the TDI of 1 mg of nickel you'd have to scrape over 11 mg of steel off your pan every day.
To reach the TDI of 21 mg of chromium you'd have to scrape 117 mg of steel off your pan every day.
A stainless steel pan weighs about a pound or 453 grams. At 117 mg per day you're losing 10% of the pan per year. Unless you're cleaning with a sandblaster, this isn't plausible.