It's all a matter of what you wish to achieve.
Some people have totally unrealistic expectations of how a hawk/axe edge should function. They expect to sharpen it like a razor today, use it hard on everything, and to hold that edge for a year past next Tuesday!!!!!! Totally unrealistic expectations!
That axe/hawk edge is a WORKING cutting edge. If you use it, it will dull. If it dulls, just sharpen it. It is such a simple concept that way too many people have forgotten or never learned.
Ditto their knives!
Before the early 1700's, many of the basic TRADE axes and "hatchets" were made from all wrought iron, with NO steel cutting edge welded in. All SOFT iron! Yes, that cutting edge dulled quickly. But it was also easy to sharpen. And after you work wrought iron for a while, you will be surprised at how well it actually does hold a cutting edge. Just ask Gerry Barker about his early work forging up axes, knives, cutt-off hardies, and other sharp tools from wrought iron.
Modern "welding" iron/steel generally has around 18 to 20 points of carbon in it - designated 1018 or 1020 (now also called A36 I beleave). So it already has more carbon in it that that traditional wrought iron, and can be "hardened" more for cutting edges. 1045 has 45 points of carbon. Many knive makers use 1095 carbon steel - with 95 points of carbon. This will harden and temper very well for cutting edges. I make most of my flint strikers out of new 1095 carbon steel.
Old files can vary a lot in their actual metal alloy content. And a bunch of modern files are made from soft low-carbon iron and have the outside teeth layer Case-Hardened. So only a small fraction of an inch of the outside of the file has high carbon content, and the inside will not harden well for a cutting edge by modern standards. So you have to really check out those files to know what you are working with.
One of the cheapest available GOOD scrounged steel for welding into your axe/hatchet for a cutting edge would be --- lawn mower blades. They are generally made from 1084 carbon steel, or 5160 alloy steel. These are TOUGH steel to begin with, since a lawn mower blade takes a lot of abuse - and generally without breaking. And they hold a cutting edge pretty well. If you search around, you can generally pick them up for free, run around 2 1/2 to 3 inches wide, 1/8 to 3/16 inch thick with some running 1/4 inch thick. The hardest part is cutting a lawn mower blade down to the size you need for your project. The rest is just a forge-welding and heat-treating project. If you want to, you could make the whole hawk head from half of a lawn mower blade. Just harden only the cutting edge.
But you also might want to ... play around ... with some wrought iron. It's an interesting project. I've made three small axes and a half dozen knives so far - simple basic "trade" items, nothing fancy. And, as noted above, they hold an edge surprisingly well for being just wrought iron.
Hope some of these humble ramblings help.
Mikey - yee ol' grumpy German blacksmith out in the Hinterlands