No Saxon invasion?

[Rumo]

There is no claim that the findings explain the observed data in unequivocal terms.

Many theories (even in physics!) do not explain the observed data in "unequivocal terms". There always are (or can be) several theories. There's no "unequivocal evidence" for the Big Bang, but it would be crazy to rant about not having 'proofs' for Big Bang! It it as proven as it can be, given our state of knowledge.

All the models used assume exponential growth rates with various plausible values (personal correspondence).

One only has to read their paper: "Population growth was modeled as an exponential from an initially constant effective population size. (p. 1011)

He didn't offer it as final proof.

"Final" is a red herring here. Halsall did not write about final/definitive proofs, but about "can 'prove'".

No they don't. They test three models which are explained in the text along with the values for the factors.

(1) Island model (TS 5),
(2) Neolithic (TS 5 240 generations BP or 6,000 years BP assuming 25 years per generation)
(3) Anglo - Saxon (TS 5 60 generations BP or 1,500 years BP assuming 25 years per generation).

Yes, they do. It's either no migration (island model, TS = infinite) or one big migration (TS = 240 generations for 6000 years BP, or 60 generations for 1500 years BP). This dichotomy is just wishful thinking and Halsall is correct to point it out.
Just read their paper:

"A program was written to simulate the coalescent under growth from an initially constant effective population size, allowing a single split TS generations ago of the parent population into two descendent populations, A and B, and subsequent background migration at a rate m, where a proportion m migrates from population A to B and simultaneously from B to A in each generation. A further extension to the program allowed for a single unidirectional migration event from B to A at a time TF generations ago, such that lineages in A immediately after this event had a probability F of having just migrated from B." (p. 1012)

So they assume a single one-directional split! One big migration! Their three scenarios (three TS-es) remind me of the positivist fallacy, the only possible migrations are the migrations they can imagine based on some other sources. A verosimile estimation should account for more dynamic scenarios, with a variable number of significant migrations in both directions.

I didn't say Halsall challenged it on technical grounds. It has never been challenged on technical grounds. It has only been criticised by people without technical expertise.

If the paper was not challenged on technical grounds, then the question of expertise is moot.
The real problem is Weale et al have no expertise in history, archaeology, linguistics, social and cultural anthropology but that didn't stop them to write a paper about Anglo-Saxon migrations. Who were the Anglo-Saxons?

OK you disagree but to me, including personal opinion like "quite apart from the fact that it seems to me to be principally aimed at getting the researcher in question into the headlines" suggests an unnecessary emotional content which betrays an underlying motivation.

"it seems" is a matter of opinion, not emotion.

So, you see for example statements such as 'the history of the Goths starts in 238 AD'. The question, 'what were they doing in 230 AD and where were they?' is unanswered but, in the wider field of settlement history, it is the question most of us want an answer to.

But it is a meaningless and ultimately unanswerable question (for archaeology - but that's a different topic, and genetics). Perhaps a Goth had an Alan father and a Thracian mother. He might even have presented himself in different circumstances as Goth, Alan or Roman. So would his DNA be 'Gothic', 'Alanic' or 'Roman'? Moreover there might be migrations undocumented in sources, and migrations in sources which were not real migrations, but the emergence of new economic, social, cultural structures and identities. If geneticists only try to prove/disprove scenarios attested in sources, then genetics holds no answers. As Halsall noted, they'll just drag us back 150 years ago, when Goths and Saxons were seen as biologically homogenous populations.

It's complete rubbish. If publicity was the motivation, they wouldn't be publishing in Molecular Biology and Evolution. It's hardly a best seller at WH Smiths.

Why would they publish on a topic outside their expertise, if not to draw attention? The other alternative I can think of is really unflattering.
It's undeniable this paper received more attention than other papers. Catherine Hills also discussed it (and dismissed it) in The Origins of the English (2003), p. 65-71