Assuredly not. At one point, the deuterostomes should have all been similar worm-like creatures, and this is the general form you see in some more basal lines like graptolites and amphioxus. But by the Cambrian, the echinoderms already their own distinct lineage, with mostly flattened or vaguely crinoid-like fossils (I mentioned Gogia, and this pdf shows some others).
Sea cucumbers are an exception to this form, but while there is uncertainty over their placement, it’s in how they relate to other eleutherozoans - sea urchins, starfish, and brittle stars. Molecular trees agree with morphology that they group with them separate from crinoids. So considering them as a model for the first echinoderms, in contrast to the earlier fossils and more basal groups, would seriously limit plausibility of a model on its own.
There are likely some Cambrian groups that branch between the modern deuterostome phyla. Some that have been invoked are the shelly carpoids and the Vetulicolia, which are vaguely larvacean-like but very ambiguous. These could count as remnants of a “same-thing” period and even show some overlapping expression, though the evidence would be poor and invoking that is another weakness, because it’s an easy excuse for any bad phylogeny. At any rate, though, by then the echinoderms themselves have moved on.
