[General Questions and Answers] burugato:
since no one has been able to give a solution and I HATE finding dead threads with OP that was able to fix but didn't describe how to anyone, I leave you MY solution here (I managed to install bromite webview)

You can still suggest better solutions if you know any

PREMISES:

I am not responsible if following these steps do not get the desired result, please backup everything before trying!

By doing this you will probably

lose safetynet pass

, in my case all of my apps that need safetynet pass are checking it only on first installation so I simply do this after installing them and while having a backup of them made with titanium

I've tested this in lineageOS 17 with TWRP recovery and root, I cannot guarantee it will work in other cases as well

What do you need:

ON DEVICE:

rooted device

magisk (may not be needed if you find a different way of making an app system app)

recovery that let you move files inside /system partition (In my case TWRP)

ON PC:

linux-based system (I think you can do the same thing even on windows or macos but you need to do extra steps in order to install tons of missing commands)

adb installed

aapt installed

java installed (seems like both oracle and openjdk are good enough), for debian-based system you can use sudo apt install default-jdk

PART 1 - ON YOUR PC:

install apktool, istructions here https://ibotpeaches.github.io/Apktool/install/

move apktool in /usr/local/bin in order to be able to call it with apktool command

(these 2 commands are launched from the download folder)

sudo mv apktool /usr/local/bin/apktool

sudo mv apktool_VERSION.jar /usr/local/bin/apktool.jar

attach device to pc via usb, activate usb debug mode in order to be able to use adb

to avoid mixing files with other things it's better to make a custom folder for the next sequence of operations, but do as you like (this is just a suggestion, if you want to do it differently just adjust the commands accordingly):

mkdir ~/bromite_ww_framework_res && cd ~/bromite_ww_framework_res

pull your device "framework-res.apk" (KEEP it as BACKUP to restore in case you do something wrong!)

adb pull /system/framework/framework-res.apk

cp framework-res.apk framework-res_ORIGINAL_BACKUP.apk

decode your "framework-res.apk" using apktool

apktool if framework-res.apk

apktool d framework-res.apk

edit the decoded "config_webview_packages.xml" and add Bromite WbView signature

(you can use the editor you like, in my example there is nano to avoid people crying about not being able to exit vim)

nano framework-res/res/xml/config_webview_packages.xml

BE VERY CAREFUL TO PASTE THINGS CORECTLY IF YOU DON'T LIKE BOOTLOOPS!

<webviewprovider availableByDefault="true" description="Bromite WebView" packageName="org.bromite.webview">
<signature>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</signature>
</webviewprovider>

rebuild the apk with the modified "config_webview_packages.xml"

apktool b framework-res --use-aapt2

create a simple apk from the build folder generated by apktool

cd framework-res/build/apk && \
zip -r ../../../framework-res_EDITED_BROMITE.apk * && \
cd ../../..

check that the new apk has the correct config_webview_packages.xml (search 'org.bromite.webview' in the output)

aapt d xmltree framework-res_EDITED_BROMITE.apk res/xml/config_webview_packages.xml | grep bromite

replace new `config_webview_packages.xml` in the original apk (you DON'T want to use the new apk! just replace the config_webview_packages.xml file in the original one!)

mkdir res && \
mkdir res/xml && \
cp framework-res/build/apk/res/xml/config_webview_packages.xml res/xml/config_webview_packages.xml && \
zip -u framework-res.apk res/xml/config_webview_packages.xml && \
rm -rf res

(if you have trouble running those last commands just do it manually via 7-zip or similar programs)

place the modified `framework-res.apk` in the device

PART 2 - ON DEVICE:

Reboot device in recovery

Mount /system partition

Move the new framework-res.apk you generated to replace the old one in /system_root/system/framework/framework-res.apk

I've wiped dalvik cache (I don't know if it's needed, but just in case...)

Reboot device and check if all is still working

If everything is ok you can proceed, otherwise return to recovery and just put the original `framework-res.apk` again

Install Bromite WebView apk from official site https://www.bromite.org/system_web_view

Now you need to make Bromite WebView system apps, to do it I've installed the magisk module app-systemizer and rebooted device

https://xdaforums.com/t/module-terminal-app-systemizer-v17-3-1.3585851/

On pc with adb shell or in terminal emulator you can launch su -> systemize

and following the instructions you can move the bromite webview in system/apps

I rebooted and everything works, the bromite webview is selectable from the developer settings 🥳

In the meantime, an update of the webview has also been published and installing it like any other apk it applies correctly 🥳

thanks to lost101 for inspiring part of this guide with this other guide https://xdaforums.com/t/tutorial-ho...ut-breaking-animations-in-stock-roms.3455024/ (I made some changes to its steps but the idea behind it is the same)