Could a Dyson fan scale up to be used as a bladeless aircraft engine?
No. Not a useful propulsion engine.

The first problem is power. The air stream from Dyson's fans is weaker than what you can get from a conventional fan the same size, and jet engines need a very powerful stream. You'll notice they're quite heavy as well; Dyson AM-06 has a thrust-to-weight ratio of just 0.06, 100 times less than a jet engine.

The second reason is they're very inefficient at high speeds. Dyson's fans use the Venturi effect, which is also used in aircraft evacuation slides, to turn a narrow stream of fast air into a wide stream of slow air. This happens through friction simply pulling slow air alongside the fast stream.

Current airliner engines do the same thing, but better. There, high-energy air spins a turbine, which then spins a fan. The conversion efficiency of this process is 3x-5x better and produces uniform velocity - important when you're close to the speed of sound. The fan approach also results in a smaller, lighter engine. 

Jet engines imitating Dyson's fans would need cores and core intakes as least the same size as current ones. The reason Dyson chooses to protect their intake with a grate and Rolls-Royce or GE don't is Dyson fans produce on the order of 50 W of power, jet engines 50+ MW. No one cares about a few watts, they do about a few megawatts, which translate into a couple ocean tankers full of oil over a plane's lifetime.

If you want, you can protect a jet's intake with a grate too - see F-15SE, Su-57 (PAK-FA) with radar blockers, Mig-29's "dirt runway" top intakes. This has nothing to do with using an ejector or not. Dyson's design does remove the slow fan, but at the cost of a lot of weight and efficiency, and it's core damage that is the worst scenario in bird strikes.

The third reason is that Dyson's fans would not scale. To fly at Mach 0.8, the ejector stream would have to be highly supersonic, which is a problem in itself. To use them for hovering, where they are reasonably efficient, you'd face the problem of Venturi effect fading with distance - requiring an entire grate of ejectors, not one simple ring. Helicopter blades do the same thing better, while being far lighter and easier to fold.