This is as much a question of metallurgy as it is of history. Note, that when I'm talking about iron I don't mean cast iron but iron with basically no carbon in it.

Making steel with the kind of furnaces they used in Europe at the time was difficult. (1) Contrary to the Indian and Chinese designs (2) of the time European furnaces couldn't melt iron fully, so there was no way to easily control the carbon content in a consistent or homogeneous way.

For steel it has to be in a very specific range but the raw iron has too much of it and it's difficult to remove while not removing too much. Some steel is usually produced more or less accidental and is easily identified because it's a slightly different colour due to carbon content.

To work against that there are techniques like case hardening were additional carbon is introduced in the surface layer at the end.

That of cause makes it difficult for archeology, if the surface layer is gone due to rust like it is basically always, there is no way to know if the iron object was case hardened. For edge retention a few tens of millimetre are more then enough of a high carbon steel layer. A soft iron/mild steel core or back can have beneficial properties and reduces the chance of snapping like seen in the traditional katana. Axeheads were often constructed with a welded on edge of high carbon steel onto a iron or mild steel body into the modern age.

However using iron or very soft steel seems common in ancient Europe, there is a reverence in comentarii de bello gallico to legionaries straightening out their swords. A medival sword would have snapped before significant plastic deformation ( if it was of a decent quality, they too had cheap materials).

You have also to consider that metals are quite easily reused and that the total amount of iron available increased over time.

(1) bloom furnaces produce something called sponge iron were the slag is melted away and the iron oxide is reduced to iron but the iron didn't melt. It produces iron with between 0.2 and about 1.5% carbon, with most of it in the low range. With a lot of work that can be welded together into a solid piece of iron.

Blast furnace, theory and practice, American Institute of mining, metallurgical, and petroleum engineers, Gordon and Breach science 1969

2 they invented blast furnaces and used them from at least 5th century BC.

"The Earliest Use of Iron in China" by Donald B. Wagner in Metals in Antiquity, by Suzanne M. M. Young, A. Mark Pollard, Paul Budd and Robert A. Ixer (BAR International Series, 792), Oxford: Archaeopress, 1999

There is a transition towards using more steel in weapons and tools in Europe, from Antiquity through to the post-Medieval period. However, it's a fairly slow transition.

Pleiner (Pleiner, R., The Celtic Sword, Oxford, 1993) examined a sizable group of Celtic swords, dated to about 400BC to 200BC. About 40% were iron (considering carbon contents below about 0.3% as iron), and 60% were iron-steel composites. Generally, they were not quench-hardened. While the ideal pattern-welded blade has steel edges, quite a few of the iron-steel Celtic swords had iron edges. Since these blades weren't quench-hardened, this doesn't make as much difference as it would if they were hardened.

Moving into Roman times, the proportion of swords (and daggers) that are iron-steel composites increases, and quench-hardening becomes common (but about half of the swords are unhardened). By late Roman times, both within the Empire and outside, most swords were iron-steel composites. For example, 90% of the Nydam swords showed decorative pattern-welding - this is the earliest known use of fancy pattern-welding in Europe (earlier swords used simple laminated structures). Some of the Nydam swords have soft iron edges (and such swords continue to be found into Medieval times).

As a sample of early Medieval swords, the Ulfberht Viking-Age swords have been extensively studied (because some of them appear to be made of imported(?) crucible steel; at least some of them are all-steel). Of 55 Ulfberht swords that were studied, 5 were all-steel, 19 were iron-steel composites that had been hardened, 16 were iron-steel composites that hadn't been hardened, and 11 were all-iron.

Williams (2012) gives the details for the swords above. He also describes some later Medieval swords, and post Medieval swords. Of 44 swords from 1000-1500, 13 were all steel and hardened, 1 was all steel and unhardened, 25 were iron-steel composites and hardened, and 5 were iron-steel composites which were not hardened.

For 16th and 17th century swords given by Williams, 5 were all-steel and hardened, and 3 were iron-steel composites and hardened.

From these samples, we see that the fraction of swords made of iron decreased, and the fraction of swords made from steel increased. However, from 400BC to 1500AD, most swords were iron-steel composites. The frequency of quench-hardening of blades increased greatly, from zero to most (the latest unhardened sword listed by Williams is from the 15th century).

The popularity of iron-steel composites for swordmaking isn't just due to the higher cost of steel compared to iron - it also allows more reliably heat-treatment of swords. Modern heat-treatment of swords usually consists of two steps: (1) quenching to harden the blade, and (2) tempering to reduce brittleness. Tempering as a separate step appears to be a relatively recent development; Medieval swords appear to have been tempered by slack-quenching (or auto-tempering, as it sometimes called). The blade is quenched, but removed from the quenching liquid before it has completely cooled down. The retained heat results in tempering of the blade. Slack-quenching tends to be imprecise, and can easily result in a brittle blade. An iron-steel composite, where the iron will stay soft (and tough), adds a significant margin of safety - even if the steel edge is brittle, the blade can remain in one piece if the edge cracks.

Reference:

Williams, Alan, Sword and the Crucible: A History of the Metallurgy of European Swords up to the 16th Century, Brill, 2012.