Nice theory you got there, shame it doesn't bear out in practice. In normal labour conditions workers have no leverage over employers to demand higher wages because workers have to take any job available in order to avoid homelessness and hunger. Workers demand for jobs is inelastic. Only in labour shortage scenarios workers have leverage to demand higher wages. But even then industry consolidation counteracts that because large corporations not only have price setting market power in the consumer market but also in the labour market. This is why the only consistent way to create similar leverage for worker in the labour market is unionization. Wages have never grown significantly enough to chip away wealth inequality except in the presence of strong and wide unionization. This is the main reason why increased labour productivity decoupled from wage increases since the 70s and 80s, when union busting started picking up around the world.

Rapidly increasing labour supply can make wages worse but that's not the main driver as there have been periods where wages have both been increasing with significant immigration and also others where wages were stagnant without significant immigration.