Some immigration is an economic necessity, which is why the Tory government allowed millions in with visas. The boat people were a very convenient distraction, as it encouraged people to focus on that more than the multiple times larger amount of legal, approved immigration.
What is needed is for politicians of all parties to be honest. They need to explain publicly why we need legal immigration and set out the consequences of not having it. One consequence would be a burning need to train more of our own people at significant cost, but I suspect that's only the start of it.
The problem is that politicians are mealy-mouthed. They know explanations would be complex and would fly over most people's heads. So instead they insult our intelligence with silly three-word slogans and imply that they're trying to stop something they have no intention of stopping.
The asylum issue is separate and has no easy solution short of becoming a pariah state. The only (partial) solution is via international cooperation, largely with the EU. That was a dirty word under the Tories and there are still many people who see any agreement with the EU as a 'sell out'. Until this juvenile attitude is fucked off we shall make no progress.
Labour will never debate this issue, because to do so they would have to address nativism, in so doing they'd have to explicitly state their view that nativism is racism.
You state....
"a burning need to train more of our own people"
Labour doesn't believe in the concept of "our own people".
That's why the right refer to the likes of Blair, Brown and Starmer as globalists because there's a sliver of truth to it, it's why Starmer feels the need to fly the Union flag all the time because he knows the left has lost the argument on nativism by default, because it's an argument they can't have, because Labour doesn't believe in nativism but can't publicly admit it.
This dissonance on Britishness led to Brexit, to the racist riots, to the general powerlessness and unease we all feel about the state of our country, our history, who we are, what we stand for.
We all blame the likes of Farage and Robinson for this mess and they are culpable grifters and agitators, of that there is no doubt, but they're a symptom, not the cause.