Seeing how transition would not happen in a moment but rather over decades, the problem would kind of solve it self by simply diminishing the amount of meat we consume from actual livestock and trend more towards vat-grown meat. Slowly, steadily, vat-grown meat production ramps up and becomes more profitable and sustainable (and as people start to trust it more), however real meat will never stop being produced. It is just like we have artificial sweeteners and actual sugar. We have GMO foods and non-GMO foods. There is market for everything.
Lets draw a parallel with our own world and how it is dealing with transitioning from fossil fuel to sustainable green energy. It is not happening over night - fossil fuel is still used very much but over time as more green-energy power plants are created less people will have a need for fossil fuels.
In your scenario people would also slowly start to transition and as there are less customers for actual meat it would be produced less as farms start to fail due to not being profitable. Some farms would still function as high-end brands that deliver 'organic meat' if that thing is allowed in the world for those that prefer it to vat-grown stuff.
But overall, decades in after vat-grown meat appears on the market it will simply push out the organic meat as farms go out of business or switch their factories to produce vat-grown instead of actual meat. Eventually an equilibrium will be reached on the market.
Animals that were grown in those farms would not go extinct - cows, sheep, goat etc. still live in nature even though we grow them for consumption ourselves.
Their numbers would probably dwindle but i doubt they would go extinct.
Plus, animals are grown (and/or slaughtered) for reasons other than meat - production of milk (and from that other dairy products), skins, fertilizer, pets...
As for controversy there will always be a camp of people that will prefer natural meat and will campaign against vat-grown stuff for the same reason people protest GMO foods today. There will be people that will protest meat eaters, just like vegans do today. There will be those that will complain about long-term health effects that we may not know. People will probably argue that vat-meat factories pollute the world more than natural farms and waste more energy. Naturally you will have politicians claiming that the industry is bad because it is taking away jobs from farmers, veterinarians, cattle growers and medicine makers that ensure cattle remains health.
Just look at any industry that has pushed some other older technology out of the market and see the arguments that people used.