So it appears that this actually is the intended behavior – the browser will default to the method you select when initially asked:
Taken from this issue here: https://github.com/brave/brave-browser/issues/7503
The options for the drop down menu will be:
Crypto Wallets , and
MetaMask if it is installed. The default will be
Ask until an explicit option is set. The new preference will be stored with the pref name of
brave.wallet.web3_provider in per-profile prefs which are visible in chrome://prefs-internals/. If the user explicitly uses Brave wallet first and opts in and the setting is set to Brave, then even installing MetaMask later won’t change the setting from Brave.
Ask is selected, UI will be shown as per the section on Infobar work
None is selected, the entire extension will always be lazy loaded, even if it is installed.
Crypto Wallets is selected, only Crypto Wallets will be used as a web3 provider.
MetaMask is selected, only MetaMask will be used as a web3 provider.
If MetaMask is not installed, the existing UX will continue to be valid:
If MetaMask is installed, the existing infobar UX highlighted in the Objective section will be repurposed to the following.
Closing the infobar will use neither provider, it will keep the setting mentioned in the last section to “Ask”.
If the user’s setting is “None”, no infobar will be shown.