How to update Android System WebView on MIUI I'm on rooted MIUI 10 8.8.9 Android 8.1

Currently there is WebView version 62.0.3202.84 installed inside /system/app/WebViewGoogle/WebViewGoogle.apk with /oat/arm and /oat/arm64 directories containing *.odex and *.vdex files

I've tried to just install newer *.apk, but I get following error "Can't install system applications from unofficial channels".
Looks like MIUI's Package Installer is blocking updates of system apps, so you have to disable "Turn on MIUI optimization" in "Developer options". After that system starts using Android's Package Installer which behaves as expected and allows to update.

Note that if you want to reenable "Turn on MIUI optimization" option, you'll have to reboot your device after that for everything to work again (e.g. your device may stop receiving sms from short numbers).