I can imagine a device that listens on a ham frequency to record voice messages and decode commands, and then retransmits recorded voice messages on the same frequency on command. Such a device would be relatively easy to design and build. But there are legal issues, at least in the US.
Legally speaking, I presume that you're talking about an automated system that can transmit without being constantly monitored by a control operator. I can think of three types of stations in which an automated system can transmit legally on the ham bands under US law (not including earth stations, space stations, and telecommand stations, which have to do with stations in orbit): propagation beacons, automatically-controlled digital stations, and repeaters under automatic control. You ruled out digital modes, and propagation beacons aren't allowed to transmit voice messages, so as I see it, the only legal way to operate such a device in the US would be as a repeater under automatic control. (Please know that I'm not a lawyer.)
Unfortunately, your device has now gotten quite a bit more complicated: in addition to listening for messages and commands, and retransmitting stored messages and acknowledgements, now your device has to also retransmit everything it hears on another frequency, and also comply with other legal requirements for automatic control such as transmitting its call sign periodically, timing out after someone transmits through the repeater for more than a few minutes, and monitoring a separate control channel (another frequency, or a telephone line, or an internet connection) for commands.
Repeaters are quite expensive, if your goal is simply to store and forward messages. But if you have a repeater already, and the repeater controller is a computer, then installing some software that works with your repeater hardware sounds like a reasonable way to accomplish what you're asking for.
The poster of the question has pointed out that there is a device marketed as a "simplex repeater" that connects to a radio and does what the poster of the question is asking for: it records messages and retransmits them later upon command on the same frequency. Unfortunately that device doesn't comply with the definition of a repeater in 47 CFR § 97.3 (I added the bold face):
(40) Repeater. An amateur station that simultaneously retransmits the transmission of another amateur station on a different channel or channels.
Because that device marketed as a simplex repeater doesn't simultaneously retransmit on a different channel, legally it's not a repeater, and therefore such a device can't legally transmit under automatic control. But the FCC has changed quite a bit from the old days, and their attention is on spectrum auctions, mobile phones, and broadband internet far more than on the amateur service, so unless they get quite a few complaints, in my opinion they are unlikely to act to prevent "simplex repeaters" from being sold.