I don't know about local send specifically, but KDE Connect will do that. And if you have an FTP client on your phone, then yes you can easily spin up an FTP server on your local network and transfer files that way

I mean if the game is in standard files on the phone... sftp server is more or or less enabled by default, or 1-2 commands to turn on in most linux distributions ("sudo systemctl enable sshd --now" would do it on distro's that don't have it on by default). and plenty of sftp clients available for ios and android. I've never used localsend, but it's pretty explicitly cross platform, snap is default on ubuntu based ones so in their examples it would install from the single command "snap install localsend".

Now running the game assuming it's a made for windows one, just requires wine to be installed. Which is default on some distro's, or a checkbox or single command to install on others, but unlikely to be a huge challenge.