I digged deeper into this issue and found a solution - but not a completely satisfying one.
First of all, when I started searching for any reason for this behaviour, I found those lines in my /var/log/messages
Jan 31 17:59:18 batocera user.warn kernel: [ 23.771862] hid-generic 0005:045E:02E0.0005: unknown main item tag 0×0
Jan 31 17:59:18 batocera user.info kernel: [ 23.772253] input: 8Bitdo SF30 Pro as /devices/platform/soc/ffe09000.usb/ff500000.usb/xhci-hcd.0.auto/usb1/1-1/1-1.2/1-1.2:1.0/bluetooth/hci0/hci0:70/0005:045E:02E0.0005/input/input13
Jan 31 17:59:18 batocera user.info kernel: [ 23.772500] hid-generic 0005:045E:02E0.0005: input,hidraw2: BLUETOOTH HID v9.03 Gamepad [8Bitdo SF30 Pro] on 00:1a:7d:da:71:13
Jan 31 17:59:18 batocera user.warn kernel: [ 23.863607] microsoft 0005:045E:02E0.0005: unknown main item tag 0×0
Jan 31 17:59:18 batocera user.info kernel: [ 23.863872] input: 8Bitdo SF30 Pro as /devices/platform/soc/ffe09000.usb/ff500000.usb/xhci-hcd.0.auto/usb1/1-1/1-1.2/1-1.2:1.0/bluetooth/hci0/hci0:70/0005:045E:02E0.0005/input/input14
Jan 31 17:59:18 batocera user.info kernel: [ 23.864100] microsoft 0005:045E:02E0.0005: input,hidraw2: BLUETOOTH HID v9.03 Gamepad [8Bitdo SF30 Pro] on 00:1a:7d:da:71:13
Apparently, there are 2 devices initiated when I pair the 8BitDo SN30Pro to the Odroid N2, and especially the second one, talking about “Microsoft” looked kind of weird.
Taking a closer look into the manual of the controller showed that there are 4 different operating modes offered: Nintendo Switch mode, Android mode, Windows mode and Mac OS mode. If you want to use one of those modes, you switch into one of those modes by pressing Start+A/B/X or Y first and then start the pairing.
You get the message above, when the controller has been started and paired in Windows mode. And here, you get this huge delay when starting titles and a delayed or non-working auto connect on a boot etc.
When you start the controller in Mac OS mode, it does not seem to work at all. Maybe configuring the buttons would have helped, but I dug not deeper into this.
Because finally, using the Switch or Android mode works without the delays and without auto-connection issues when starting Batocera. But I discovered at least two drawbacks for now: Vibration does not work here (tested with Gran Turismo 2; works in Windows mode), and the Home key also does not seem to work (also works in Windows mode).