Short answer.

• For ordinary tangential structures (X, ζ) the cobordism hypothesis does not give anything “new”: one has a canonical identification |Bordn(X,ζ)| ≃ Ω∞−n Mζ, i.e. the geometric realization of the n-dimensional (∞,n)-bordism category is the (∞−n)-fold loop space of the Thom spectrum Mζ. This is the same identification one gets from Pontryagin–Thom/“GMTW” results; it is not a different universal characterization.

• For manifolds with Baas–Sullivan singularities, Lurie’s universal property (Theorem 4.3.11) says that BordnZ is the free symmetric monoidal (∞,n)-category with duals generated by a “Z-point”. After geometric realization and group-completion, one again obtains the 0th space of the (connective) spectrum represented by the corresponding Baas–Sullivan Thom spectrum MZ: |BordnZ| ≃ Ω∞−n MZ. In particular, this recovers the familiar connective “quotient” spectra MU/I (killing a regular sequence), and hence the connective chromatic truncations like BP⟨n⟩, obtained by killing v>n in BP∗ (via their preimages in MU∗).

• You should not expect to characterize spectra like BP, E(n), or K(n) as geometric realizations of bordism (∞,n)-categories via “preservation of colimits”. There are two basic obstructions:

  1. Connectivity: |Bordn(–)| always produces the (∞−n)-space of a connective spectrum. Periodic theories such as E(n) and K(n) are nonconnective, so they cannot arise this way (at best one could obtain connective covers or connective approximations, e.g. BP⟨n⟩).
  2. Thom nature: The cobordism (∞,n)-categories Lurie constructs give Thom-type spectra (ordinary Thom or Baas–Sullivan Thom spectra built by successive cofibres that kill regular sequences). Many chromatic spectra are not of Thom type in this sense. In particular, BP is widely known not to be a Thom spectrum coming from an n-fold loop map; this follows from general constraints on THH of Thom spectra (see Blumberg–Cohen–Schlichtkrull, “THH of Thom spectra and the free loop space”) together with computations of THH(BP). Thus there is no BordnZ whose realization would be Ω∞−n BP. Moreover, BP itself is obtained from MU only after p-local idempotent splitting; retracts/splittings are not colimits, so the left-adjoint property of |−| does not help to isolate BP from MU.

What you can get.

• Starting from MU (or another oriented theory) and a regular sequence, Baas–Sullivan give a bordism theory with prescribed singularities whose representing spectrum is the cofiber-iterated Thom spectrum MU/I. In particular, at a fixed prime p, one can arrange I so that the resulting spectrum splits (after p-localization) into summands one of which is BP⟨n⟩, and then |BordnZ| ≃ Ω∞−n BP⟨n⟩ for the appropriate singularity datum Z. This matches the classical description and uses only that geometric realization identifies the grouplike E∞-space of the cobordism category with the 0th space of the corresponding Thom spectrum.

• Localizations/inversions (e.g. passing from BP⟨n⟩ to E(n) by inverting vn) and idempotent splittings (to obtain BP from MU) are operations in the category of spectra/E∞-ring spectra that are not produced by colimits of symmetric monoidal (∞,n)-categories in any straightforward way detected by |−|.

Conclusion. Geometric realization preserves colimits, but that fact does not yield new universal characterizations of chromatic spectra beyond the classical Thom/Baas–Sullivan picture. What the cobordism hypothesis gives is a universal property at the categorical level (free symmetric monoidal (∞,n)-category with duals and prescribed singularities). After applying |−| and group completion, one simply recovers the known identification with Ω∞−n of the corresponding (connective) Thom/Baas–Sullivan spectrum. Consequently:

• You do get connective chromatic approximations (e.g. BP⟨n⟩ and MU/I) in this way.
• You do not get E(n) or K(n), and there is no general “universal characterization” of BP via |−|; BP is not of the Thom type that arises from these bordism categories. References for these identifications and obstructions include GMTW on the homotopy type of the cobordism category, Bökstedt–Madsen on its infinite loop space structure, Lurie’s “On the classification of topological field theories” (especially §4.3), and Blumberg–Cohen–Schlichtkrull on THH of Thom spectra.