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Posting this to answer my own question in case it helps anyone else.
My Garmin was beeping so often while riding off-road, it was getting very annoying and potentially dangerous if it distracted me at the wrong moment.
"Maps do not have routable roads in this area" while following a GPX off-road. Beep, and message covering up half the map on screen that stays there, comes back in 2mins after dismissed. Still happens if recalculation is disabled. Solved by not actually using navigation; instead go to the course you want, then Settings > Always Display.
"Off course", "course found", "recalculate?/recalculating" beeps when the GPX doesn't exactly match the way on the ground, or you're in thick forest. Solved by same as previous.
Other obvious things to turn off - Jump Alerts, Segments.
Virtual partner - plays a ridiculous melody when this he finishes. Can't be turned off.
Texts and calls - can only be turned off completely, no option for just muting.
For these two reasons I turned Tones off completely in system settings. Bit of a pain if you wanted them on for road riding though.
Which model? Sounds like a newer one given: Jump alerts.
I don't think I've ever heard a peep from the virtual partner on either the 510 or 520. Might be a new "feature".
Text and calls - just turn the bloody phone off! Or at least bluetooth so the two don't talk to one another.
830.
I keep the phone on as I have OS maps on it and also use it to take photos. If I didn't want them on the Garmin at all, it would be easier to disable this completely on the Garmin activity profile, as then it's one less pre/post-ride thing to remember.