I think the version on 14.04 is too old. I found the statement you need 3.9 for that (is that somewhere documented properly?).

For what is worth, the sqlite3 version in Ubuntu 16.04 has that json1-extension loaded by default:

    sqlite> CREATE TABLE user (name TEXT, phone TEXT);
    sqlite> INSERT INTO user VALUES("tester", '{"phone":"704-56"}');
    sqlite>  SELECT DISTINCT user.name FROM user, json_each(user.phone) WHERE json_each.value LIKE '704-%';
    tester

I'm now even more happy I upgraded my server to that version yesterday, the server for that project was like yours on 14.04. It was even also because of sqlite, I wanted to have a version that supports the WITH clause. Upgrading to 16.04 (actually, I made a fresh install and moved the data over) seemed like the easiest way to get that.