The most important factor in profitability is the size of the market -- and a company's share. The top two or three producers in any market, by market share, tend to be the most profitable. Small market - you don't make many examples. Large market - you need to be a large scale producer to compete. So, potential volume is set by demand, not some arbitrary notion of bigger is better.
If a company competes is a large (by unit volume) market, then there are surely economies of scale.
Purchasing effienciency, as you know, is one. The ability to spread overheads over a larger base is another. Automation is a third - though even somewhat smaller companies can now afford things like CNC machines.
A potentially larger advantage of scale, for those aware enough to profit from it, is the entire experience curve.
Just like the more swings you take at a golf ball the better you should get, volume provides opportunity to learn and find better / faster / cheaper ways of doing things. This can include everything from more experienced workers, to more productive equipment, to redesign of the product for ease of assembly and repair.
There's a whole school of management thinking - often correct - that a company should sacrifice profits early on in order to get fastest down the learning curve. This even happens at country scale - with nations subsidizing favored industries in order to advantage their producers in gaining experience - or maybe just squeeze out competitors.
Getting fastest through an experience curve is a good thing -- at least for customers -- if a manufacturer actually improves the product and processes and passes on the cost savings.
Sometimes it leads to an early dominant market position and subsequent abuses from a single monopolists or a cabal of oligopolists. There could be a whole tangent on this. Large incumbents can not only get special tax and regulatory privilege - they can use it as a barrier to smaller competitors. Or another example, not only may they get lower prices for materials, they may lock up supply.
Bottom line - I'd concentrate less on "economies of scale" and spend more time on the factors that lead companies to be the fastest and most responsive learners in any given market. There's also an "economies of scope" argument that companies that get really good at one sort of product or process also have an advantage. So, while markets tend to be dominated by a couple of producers with broad customer appeal and high market share - there are also usually niche players with high end appeal, stripped down low cost versions, especially reliable versions, especially high performance versions, and so on. Then, there is the greater ability of small players (without so much invested in the status quo) to disrupt an industry with innovative products and processes. It didn't matter that RCA got the bits for vacuumn tubes really cheap when scrappy upstart Intel came around.
Both the economies of scale and the economies of scope arguments come back to how quickly a manufacturer learns about their customers' needs and improves their product and process to best serve them.