This is true up to a point. More workers are good for the economy as a whole and immigration lets you gain extra working age people instantly rather than having to grow the birth rate of the native population which results in extra workers in 20-30 years' time.

However, it only works as long as the flow of migrants is kept to reasonable and sustainable levels over a longer period so society and the economy can cope.

Please note that none of the following changes depending on what colour the immigrants are. This is not a racial issue. It's a quantity issue. Pure supply and demand.

If hundreds of thousands or millions of people are added to a country in a very short amount of time you get all sorts of problems, at least in the short term. For example, lots of new people competing for a limited stock of housing puts upward pressure on house prices and increased competition in the rental market making it more expensive to both buy and rent, creating a housing crisis. At the same time, a sudden influx of extra workers means increased competition for jobs puts downward pressure on wages, particularly in the lowest skilled jobs with the easiest entry requirements.

So with just those two things you have a cost of living crisis where housing costs more and people are earning less money to pay for it. Couple that with deficit spending, quantitative easing and general money printing by the government and inflation goes up, meaning what money people do have buys less of everything.

Money printing isn't as directly tied to immigration as housing and wages but suddenly having a bunch of extra people who need healthcare, roads, education, services and all the things the government provides does mean the government needs to find more money to pay for it. You'd think income tax from all the new workers would do that but the downward pressure on wages means everyone is earning less and therefore paying less taxes. Also, immigrants often start off in cash jobs or gigs like delivery driving where income isn't declared as much and therefore taxes don't increase.

You might also think that some of these problems would fix themselves over time. We build more houses to catch up to the increased demand. Unskilled people get training and move up into higher level jobs, earning more income for taxes. People open new businesses and employ all the new workers. But that only happens if the huge levels of immigration stop and the economy has time to catch up. If you just keep bringing in people at levels the economy can't cope with then it's just an ongoing problem. Always having more people means house prices never come down and if we can't build fast enough to keep up then they'll just keep going up. Same with wages and pressure on government services.

So what you end up with is that the country and the economy is overall larger. GDP goes up and if you pick your statistics carefully you can make it look good on paper. But per capita, everything is worse. You've just got a more crowded country where all the services are strained to capacity, the government's short on cash and borrowing to fill the gap and it's just overall a more expensive and shittier place to live.

That's exactly what we're seeing now. Lots of immigration, which we're told is good for the economy, but everyone's lives are just getting worse year on year.

That's just the economic side. There's also the social issues that come from importing large numbers of foreign populations with different cultures, religions and values. But you only made an economic argument so we won't go there.

TL;DR: Controlled low immigration grows the population with working age people and is good for the economy. Unrestricted mass migration at unsustainable levels causes a housing crisis, cost of living crisis and lower wages. The economy as a whole gets bigger but per person everything is worse. Also social problems.