They 'own' them in the literal sense of ownership. Cede and Co is the name recorded on the issuer's stock ledger. In case you aren't familiar, the stock ledger is something issuers are required to maintain and use to track ownership. Very commonly this responsibility is outsourced to companies called Transfer Agents, which themselves need to be SEC approved.
I would definitely encourage checking out the SEC's recently updated page on the options investors have when holding securities. It's very readable and will likely answer your questions.
TLDR - If you own shares in a broker, you are a "beneficial" owner. This means that while the economic and voting impact of ownership are supposed to be passed on to you, you are not the named owner. If you own shares directly on the register of the issuer there is no middleman to pass these things to you.
DRS is not about price impact on any security. There should never be any price impact on a security from investors choosing DRS over an alternative holding method. DRS, rather, allows for other assurances - most critical for me personally are 1. being able to submit shareholder proposals directly to the company without needing to go through other channels and 2. knowing that my votes will not only be cast, but counted. For more on 2, know that over voting is a massive issue in shareholder democracy, and companies holding elections or seeking shareholder input on proposals never get to see that. Proxy vote counting companies truncate or control voting results before reporting.
This is (imo) a fascinating and tragic problem. Here are a couple sources to get you started if you feel the same way.
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=904004
https://web.archive.org/web/20060421085925/http://www.rgm.com/articles/FalseProxies.pdf
https://katten.com/files/21384_proxy-vote-processing-issues.pdf
I agree with you completely regarding the massive improvements to liquidity and settlement with a centralized depository model. The Depository Trust was founded to that end and accomplished it well. However, I do not believe the 'control' of the shares is adequately dispersed to beneficial owners under the current system. See the concerns (long standing over decades) regarding shareholder democracy from my previous comment.