Why is the molozonide adduct thermally unstable?
Why is the molozonide adduct thermally unstable?

The molozonide adduct (1,2,3-trioxalane; the initial adduct by 1,3-dipolar cycloaddition of alkene and ozone) is thermally unstable because it contains two weak $\ce{O-O}$ bonds (Ref.1). Mechanistic studies reveal that the ozonide (1,2,4-trioxalane) is formed by a sequence of three pericyclic steps involving 1,3-dipolar cycloaddition, retro-1,3-dipolar cycloaddition, and again 1,3-dipolar cycloaddition as shown in following scheme:

Also, I'd like to point out that the initial molozonide adduct is not always symmetric. It is symmetric only if the alkene is symmetric. And, the rearrangement of 1,2,3-trioxalane to 1,2,4-trioxalane is not about the stability of $\ce{C-C}$ in 1,2,3-trioxalane as OP argued. It's about thermal stability of consecutive $\ce{O-O}$ bonds in 1,2,3-trioxalane (recall that one $\ce{O-O}$ bond in peroxy compounds is not thermally stable, e.g., $\ce{HO-OH}$).

Knowing that there are two symmetrically placed $\ce{O-O}$ bond in the adduct, the cleavage of the only one bond is weird.

The accepted mechanism (Criegee Mechanism) says one of either $\ce{O-O}$ bond in the initial adduct can be broken to give a corresponding carbonyl oxide (a dipole) and carbonyl compound (dipolarophile), which undergo second 1,3-dipolar cycloaddition to give stable ozonide (see above scheme). None of the mechanism says one $\ce{O-O}$ bond cleaves selectively. it could be either one:

However, the second 1,3-dipolar cycloaddition of either set of carbonyl oxide (a dipole) and carbonyl compound (dipolarophile) gives identical ozonide. This second dipolsar cycloaddition $([_\pi2_\mathrm{s} + _\pi\!4_\mathrm{s}])$ has also been confirmed by $\ce{^{17}O}$-NMR studies (Ref.2):

The mechanism of ozonolysis was revisited with the use of $\ce{^{17}O}$-NMR spectroscopy. In a crossover experiment with $\ce{^{17}O}$‐labelled benzaldehyde and the ozonides of styrene and ethylidenecyclohexane it was shown that only the ether bridge of the secondary ozonides is carrying the $\ce{^{17}O}$ label. This is contrary to results reported earlier and confirms the Criegee mechanism.

References:

Dipak K. Mandal, “Chapter 4: Cycloadditions 1: Perturbation Theory of Reactivity, Regioselectivity and Periselectivity,” In Pericyclic Chemistry: Orbital Mechanisms and Stereochemistry; First Edition, Elsevier Inc.: Amsterdam, Netherlands, 2018, pp. 107-190 (ISBN: 978-0-12-814958-4).
Christian Geletneky, Stefan Berger, “The Mechanism of Ozonolysis Revisited by $\ce{^{17}O}$-NMR Spectroscopy,” Eur. J. Chem. 1998, (8), 1625–1627 (DOI: https://doi.org/10.1002/(SICI)1099-0690(199808)1998:8&lt;1625::AID-EJOC1625&gt;3.0.CO;2-L)).