I disagree (though you are right to say that, on average, children with fewer siblings have better educational outcomes).
Even if the primary short-term driver for high (and increasing) net migration in the UK is to fill low paying jobs, long term, we cannot meaningfully reduce immigration without either increasing birthrates.
Japan is a good comparison. Japan has a lower birthrate than the UK but both are well below the replacement rate of 2.1 children per family.
The UK's population has continued to grow, despite its low birth rate, because of immigration (with net zero immigration, our population would have been dropping for many decades now).
Japan opted against high immigration, so has experienced significant population decline (and, if trends continue, the population of Japan will halve between now and the end of the century).
Both countries have aging populations but the problem is even more acute in Japan. Not only is Japan's population shrinking but it is aging (as is the UK's), meaning that the proportion of working age people is reducing, and the number of retired people increasing. You cannot sustain a society (certainly not one with a state pension or with affordable healthcare) if there are ever-fewer taxpayers.
Japan is now investing billions of pounds in financial incentives to try and increase birth rates, by making it easier for people to afford to have children, to try to prevent societal collapse (without resorting to immigration).
Making it more affordable for people to have families will not, in and of itself, solve population decline but it is a necessary step.
Unless people in the UK start having more children, we will need to increase immigration, not cut it.
Personally, I don't really mind which way we go (though I would like to see child poverty addressed regardless) but I think people who want to cut immigration need to accept that this will necessitate significantly boosting birth rates - which is not going to happen if people cannot afford larger families.
And, again, this isn't me trying to trot out a left-wing "gotcha" argument, it's an argument being made by right-wing, anti-immigration policians (like Braverman) and is the reality for some countries (like Japan) who have had kept immigration levels low.