I'm not sure if Labour actually ever promised bringing down immigration to tens of thousand in early 2000s but I know that Labour did that apology stuff in 2015 to get more votes. But if one were to look more into it and connect some dots you'd realise that the problem was in an area different to the area of the proposed solution. Firstly, the UK will have an immigration of 10,000 a year or so, it's just impossible for a globalised western economy even after leaving the EU, let alone achieve that while being in the EU. But with that said the thing about immigration is that often the non EU immigration numbers are tied to the EU immigration numbers so anytime they want to chop some numbers off to look good they go for non EU migrants, specifically speaking it's the non EU students that get affected the most as they are the most vulnerable (and ironically the most beneficial to the economy of UK), by making it really hard for them to find work. It makes no sense that they train and educate people in shortage areas and then want the talent to leave the country.
The other thing about non EU migration is that it's almost entirely for high skill jobs that already have a shortage, they rarely if ever eat into the unskilled labour market and it's the only type of immigration that UK has full control over. So the only way Labour of early 2000s could have brought down the immigration numbers would have been by cutting on non EU immigration, which is EXACTLY what the Tories did. On the other hand the only migrants that ate into the UK unskilled labour market were the EU migrants, which the government could not possibly have prevented from entering the UK. But even then the UK had some control in that for an EU migrant to stay long term they would have had to work nad pay taxes (this is a policy that all EU countries follow actually, so no one is just coming in and sitting around reaping benefits indefinitely).
I guess what I am trying to say is that regardless of there being 100,000 or so immigrants now vs 300,000 a few years ago, barely anything changed when it comes to the unskilled labour market because they targeted a group of migrants that don;t do unskilled labour. Then there is the fact that there was an economic crisis in the 2nd half of 2000s so everyone wanted this cheap labour from EU. It was just a shitty time and even if the Tories were in power they wouldn't have been able to do anything about it as even if they had brought the numbers down (like they did in the following years) the EU migrants would still have been able to enter UK and provide cheap labour. So blaming that lack of jobs in the unskilled labour market on Labour's immigration policies is kind of a leap as nothing they or anyone could have done would have prevented that from happening during that time.