Following is a quick disassembly of a 3€ 🛒 no-name Chinese BT FM-transmitter with Bluetooth 5.4, Aux and SD-Card. Aux-port has plug-in detection an is used as line out. Input is either SD-Card or it is used as Bluetooth audio receiver and its output is either AUX or transmitted over set FM frequency - powered by USB-A plug (no batteries, nice! :)).
Next to the 7 segment display, there is a microphone for hands-free input over bluetooth.

If that is too much, there are simpler Bluetooth AUX only adapters for ~2€+
If you need more DIY, there are 🛒FM transmitter DIY solder kits on aliexpress for ~2-3€.
If you need a way to set frequency over software for a project, check out Sony Ericsson MMR-70 FM transmitter.

There are 🛒line-in cassettes (~4.50€) (even🛒with bt too for ~9.50€ for your vintage stereo).
Also see "Gadgets you can get for your car"

The module 'talks': sound when starting, short voices for 'connected' and mode selection (AFAIK it cannot be disabled in a simple way).
The clipped USB tin shoe holds the casing together well.
Powered by 5V from USB-A, which has data lines going to MCU/SoC. No output on dmesg/lsusb though :/
AUX is 3-pin TRS stereo (no mic). Microphone on PCB is connected over bluetooth.
The MCU is AB-series from Bluetrum. There is a gitee repo of bluetrum. On first sight, it does not seem to be much of use (?)
The closest SoC I could find quickly on the HP is AB5305B
Just for reference: There is a getting started tutorial for RISC-V based AB5301A.

On the PCB of the FM transmitter you can add a socket for antennas (or solder directly) (there are pads, just cut the traces behind).
Fun fact: protective plastic sheet is still on my 7 segment display (even though it is behind the casing)

Find out what file types and sample rates can be used with SD-Card.
Measure frequency of AUX output.