Recent ALMA observations unveiled the structure of CO gas in the 23 Myr-old $\beta$ Pictoris planetary system, a component that has been discovered in many similarly young debris disks. We here present ALMA CO J=2-1 observations, at an improved spectro-spatial resolution and sensitivity compared to previous CO J=3-2 observations. We find that 1) the CO clump is radially broad, favouring the resonant migration over the giant impact scenario for its dynamical origin, 2) the CO disk is vertically tilted compared to the main dust disk, at an angle consistent with the scattered light warp. We then use position-velocity diagrams to trace Keplerian radii in the orbital plane of the disk. Assuming a perfectly edge-on geometry, this shows a CO scale height increasing with radius as $R0.75$, and an electron density (derived from CO line ratios through NLTE analysis) in agreement with thermodynamical models. Furthermore, we show how observations of optically thin line ratios can solve the primordial versus secondary origin dichotomy in gas-bearing debris disks. As shown for $\beta$ Pictoris, subthermal (NLTE) CO excitation is symptomatic of H$2$ densities that are insufficient to shield CO from photodissociation over the system's lifetime. This means that replenishment from exocometary volatiles must be taking place, proving the secondary origin of the disk. In this scenario, assuming steady state production/destruction of CO gas, we derive the CO+CO$2$ ice abundance by mass in $\beta$ Pic's exocomets to be at most $\sim$6%, consistent with comets in our own Solar System and in the coeval HD181327 system.