You may want to double-check your priors on the incentive structures.
For goods and services where demand is inelastic and the barrier for entry is high, the invisible hand will end up colluding to extract the most value possible (housing, utilities, groceries, healthcare). If the state does this, the ruling party starts catching strays and is eventually replaced. Of course, for the distinction to matter, you need to be able to meaningfully create new parties.
Your point stands for elastic demand, where you can choose to vote with your wallet (restaurants, vacations, gyms). Capitalism did a good job of adding to this list during its growth stage (transportation, entertainment, communications) but nowadays it's mostly established cartels that dominate all of the above in what appears to be a race to the bottom.
The incentives for package delivery are closer to the first category than the second though, since consumers end up with whatever company the supplier signed up with, thus mostly eliminating choice and reducing quality to the bare minimum required by the contract (not unlike the postal services).
Even when the customer gets to choose, the end results are still worse than vertical integration. Consider the food delivery apps that replaced most restaurants' in-house delivery teams: they have a direct interest to collude to keep prices high since they all earn the same no matter what restaurant you buy from, whereas for the individual restaurant it makes sense to throw in free delivery to get you buying from them.
I guess the biggest downside to private ownership is that it creates an incentive to have collusion service providers masquerading as adding value (realtors, brokers, speculators, resellers, risk managers, etc.) while providing plausible deniability to the state (which is of course responsible for its failure to regulate the incentive structure for these value-added services on a case-by-case basis).