You've essentially just described the current system. Most shares on NASDAQ and NYSE etc. are technically held by Depository Trust Company via its nominee, Cede & Co. [1]. Through a complicated set of regulatory and contractual arrangements, public companies, the beneficial owners of their stock (i.e. the investor at the end of the chain) and each intermediary (banks and brokers, etc.) all maintain a sort of legal fiction that the shares are "owned" by Joe Schmoe, even though all he really has is an attenuated set of contractual rights that flow through the various intermediaries between him and "his" shares held by Cede.
Joe Schmoe does not technically or legally own those shares. Cede does. Believe it or not, you were spot on in predicting that the third party would allow "voting by proxy." That's exactly how Joe Schmoe (i.e. all of us) must vote our shares if we want to participate in a stockholder vote. We can't just show up at the meeting (or fill out the company's proxy card). You send a "voting instruction form" telling your broker how you'd like to vote, and your broker then tells Cede & Co. how to vote your shares at the stockholder meeting.[2]
To address your specific point, tenure voting would surely be based on the tenure of the beneficial owner (i.e. the person at the end of the chain who gets to vote) not the nominee holding the shares in "street name" on the beneficial owner's behalf. This might take some reworking of the arrangements between the brokers, DTC, clearinghouses, etc. (likely needing to be be built into the financial systems that log transfers and ownership, if not already provided for) but would not really pose a significant barrier.
[1] https://www.bloomberg.com/view/articles/2015-07-14/banks-for... [2] https://www.sec.gov/spotlight/proxymatters/proxy_materials.s...