Keep that Yamaha BC3A in good shape... while MIDI is MIDI is MIDI, The BC3A is not a MIDI device. (... no more than a FC3A or FC4A is a MIDI device. Although the synth will generate MIDI data when the controller is used, your Motif ES does not address itself initially with MIDI).

The Yamaha BC3A does not connect via MIDI, it connects directly to the controller and becomes one of its appendages.
True the Keyboard Controller Outputs cc2 when engaged but the BC3A does not address the Host keyboard with MIDI data, rather, the keyboard generates MIDI data in response to you moving that controller.

So what’s the difference? The difference is when you want to record yourself performing.

A “breath controller” that attaches to your system via USB-MIDI or 5-pin MIDI works to control but don’t expect to record your performing with it unless you can solve the issue of merging the note-on and CC2 data.

Your note-on still occurs via Keyboard input, and it is this info that is ‘merged’ with the BC3A input, to the MIDI OUT...

Extra Credit:
The Yamaha Breath Controller pre-dates MIDI. First available in the early 1980s on a Yamaha CS-series analog synth, and expanded the controller set (you didn’t use a sustain pedal on analog synths, but volume via Foot Control was used). The MW was also a 0-127 Control that, like the BC and FC, sends 0-127 as a controlling value. The BC added the player’s ability to control parameter change by blowing into a mouthpiece — opening another way to control and animate sound (foot, hand, breath). It does not address the host product with MIDI.

You assign it to a specific parameter (like filter Cutoff, or Element/Operator Volume, or Part Volume), the activity of ‘working’ the controller generates a stream of Control Change messages, that when recorded and played back to the synth will control that assigned parameter.

So while MIDI is MIDI is MIDI, the BC3A is not a MIDI device, per se. it is a device - much like the Sustain Pedal, or Foot Controller - that works without MIDI, but can generate MIDI for external use. If your alternate (non-Yamaha) BC connects via MIDI, you’ll need a way to merge both your note-on events (generated by pressing keys) with the stream of controller events generated by the MIDI BC... so that both are documented to your MIDI Track.

If your alternate BC connects via 5-pin you can record to the internal Performance recorder.
If your alternate BC connects via USB you must use your computer or DAW host to merge the data and route the signal.

Just so you know...