Iron production is one of my hobbies! It works depending on how high the concentration of iron is. Iron ore can sometimes come from high-purity ores like Hematite (70% iron), but it's more common to find small amounts of iron mixed in with other minerals. Red oxide clay or iron sand requires sifting and melting to extract iron from the rest of the dross. In the case of iron sand small iron filings could be sorted from the nonferrous materials with a small magnet.
That said I doubt they'll go full Dr. Stone on it and break down the process accurately. It will be macguffin iron in the end.
please dont mix iron (material, pure Fe) and iron ore (iron oxide Fe3O4 etc + other minerals/stones), otherwise those who dont know chemistry/material science will make the same mistake as the mangaka.
its true that some form of iron oxide are magnatic, so you could find them with a magnet if the iron ore has a high enough purity of the right iron oxide. but the pocket magnet isnt nearly as strong to react to such a ore.
the mistakes the mangaka makes is that he assumes pure iron can be found in nature in the first place. japans mostly got their iron from iron sand, which is magnetite (FE3O4) and therefore magnetic like pure iron (ferrit). but thats also an oxide and not pure iron which most people dont know.
and if we talk about reducing iron ore to create iron (as a material), you have two ways;
either heat it up in a carbon (C, CO) rich enviroment (oven), so that the iron oxide gets reduced to iron, without actually melting the iron/iron ore (iron oxide has a way higher melting temperature than iron at above 723°C, depending on the carbon content), which leaves you with lots of impurities/oxide which needs to be forged out => you get low carbon steel, but you can increase the carbon content while forging
or
reduce the iron as before but than use enormus heat to actually melt the iron, so that all iron oxide gets reduced. but that results in a really high carbon content in the iron which makes it pig iron, so you need to inject pure O2 into the liquid iron to get the carbon out => you are left with pure steel with a adjusted amount of carbon
which also is where dr stone made a mistake, cause they actually had shown molten iron in anime/manga, which would had a way to high carbon content (they didnt inject oxigen) to magnetice it