I'm not sure if JTAG will even work or if they connected a pin to some sensor or similar on the board. SWD will work for sure, I saw the two SWCLK and SWDIO connectors on the board.
They're a little bulky, but if you're interested in bare-metal programming, the embedded Rust ebook[1] currently uses an STM32F3 discovery kit[2] as a target, and that board's user manual has instructions for using its ST-Link to program/debug an external application[3] like your drone.
[1]: https://rust-embedded.github.io/book/
[2]: https://octopart.com/stm32f3discovery-stmicroelectronics-241...
[3]: https://www.st.com/resource/en/user_manual/dm00063382.pdf (PDF, see section 6.2 for STLink info)
You can use JTAG probes that implement the CMSIS-DAP (also called DAPLink) protocol, so you get a probe that doesn't need any driver as is seen as an HID USB device. I ran through some problems when trying to use the ST-LINK/V2 with Linux, but that was some years ago and I don't know if things have changed since then.
CMSIS-DAP: https://os.mbed.com/handbook/CMSIS-DAP
Some probes: https://www.adafruit.com/product/2764
ST actually sells an evaluation board for learning about drone control; it has a BTLE module, four small FETs, sensors for pressure and a 9-axis IMU, a 1S battery charging circuit, and source code for some example firmware on Github. Search for 'STEVAL-FCU001V1'.
I've also seen a lot of cheap STM32-based FCU boards from the same places you'd get one of these 'blue pill' boards.
These are very useful chips to learn about, because they come in all shapes and sizes. Some focus on power efficiency, others on speed, others on connectivity or signal processing, and there aren't too many differences that you need to account for in your code.
Also, you can program and debug these using OpenOCD with a generic "ST-Link" module; they only cost a few bucks from cheap sources, or many of ST's evaluation boards have a built-in STLink probe which you can use. That lets you connect to the chip from GDB and step through your code like a normal C program, which is often a lifesaver.