If I had to guess, I’d say it’s due the analogue path on the Osc input. As you said, the FX input has plenty of oomph and I remember feeding an electric guitar into a Monomachine without a DI and having a very healthy signal but there’s no way you could do it with the A4.
So, yeah, an analogue versus digital thing is what I’m wagering …gotta boost it somehow. Filter OD?