I'd also recommend an Etrex 20x.
It runs about 24h on a pair of AA rechargeable (I use black Eneloop), which is good for 300 km, or maybe 400. After that you just pop in a spare pair (at a control, unless you forget).
It works using real buttons, which work when wearing gloves, and when it's raining (which touchscreens often don't).
In common with other Garmins, it's tough and reliably waterproof without the need for a case or bag.
Compared to the "Edge " cycling models, it's bulkier, doesn't have the same variety of neat mounts available, and doesn't do many "training" type functions, just concentrating on navigation.
The official handlebar mount is sound, if somewhat on the crude side, but the cheapo ebay versions have a reputation for not being very secure (or at least used to).
For longer rides (which audax rides usually are), anything with a built-in battery is likely to need supplementing with an external battery.
This is untidy, and USB plugs & sockets are neither particularly water resistant nor mechanically robust, being designed for use somewhere like the bedside table. It's not common, but a dead USB socket on your GPS or phone would be very annoying, and possibly expensive.