The reason I chose a crystal filter is that they tend to be narrow band and have very steep skirts, so showing frequency shifts like this is demonstrated much more clearly.
For a relatively gentle LC filter, even of several orders, you'd be hard pushed to show a 1.5kHz deviation at 10MHz. Indeed, for fiddling with LC filters at HF or above, I'm happy to be educated otherwise, but I wouldn't think the frequency shift thing is much of a problem practically speaking, a gentle breeze on the coil probably might have more effect.
For tuning a narrow band IF stage though, this is definitely going to be an issue, and I'd hazard a guess that at its price point, many of the purchasers of the DSA815-TG are ham radio operators who'd use it for just this kind of thing.