The double frequency you used here ('often', 'people') made me wonder if this is local folk wisdom/ old timers' tales / [insert various other labels for this category] rather than something you happened across on the internet. You Greeks have been raising chickens since at least the 7th century BC, and Aristotle had nailed a lot of the incubation story (with outstanding observational skills) already in the 4th century BC, so I've been digging a bit, and guess what? I really think they may be onto something.
Our received wisdom is that incubation does not begin until the end of the egg-laying period to ensure synchronous hatching of the clutch of eggs. But there's a problem with this.
When a hen goes to lay the second egg in the clutch, she is unavoidably starting to incubate the first egg laid. I have read nothing to date (and I've read a lot

) to indicate that there is some physical or biological mechanism in the egg that can distinguish between this short incubation period and incubation proper. After dropping the second egg and sitting on it (and egg 1) till the bloom's dried, she stops sitting/incubating, and goes about her daily business, till she comes back the next day to lay the third one, and thereby temporarily incubates egg 1 again, and now egg 2 as well. And so it repeats until she deems the clutch complete, by which time egg 1 might have been partially incubated say 6, or 9, or 12 times. I think it quite possible that the earliest laid eggs in the clutch start to develop and then die early through this stop-start process of growing the clutch. It would be relatively trivial to test this hypothesis.
Infertility/early embryo mortality is an active new research area, but no-one I've read so far has suggested what I've just said as a potential mechanism. They do however identify lots of other things that may be impacting fertility/early mortality rates. The interested might want to see e.g. Hemmings N, Evans S. 2020 Unhatched eggs represent the invisible fraction in two wild bird populations. Biol. Lett. 16:20190763.
http://dx.doi.org/10.1098/rsbl.2019.0763 (open access)
Abstract: "Prenatal mortality is typically overlooked in population studies, which biases evolutionary inference by confounding selection and inheritance. Birds represent an opportunity to include this ‘invisible fraction’ if each egg contains a zygote, but whether hatching failure is caused by fertilization failure versus prenatal mortality is largely unknown. We quantified fertilization failure rates in two bird species that are popular systems for studying evolutionary dynamics and found that overwhelming majorities (99.9%) of laid eggs were fertilized. These systems thus present opportunities to eliminate the invisible fraction from life-history data."
and/or see Assersohn, K., Marshall, A.F., Morland, F. et al. (2 more authors) (2021) Why do eggs fail? Causes of hatching failure in threatened populations and consequences for conservation. Animal Conservation, 24 (4). pp. 540-551. ISSN 1367-9430 Open access version here: White Rose Research Online
https://eprints.whiterose.ac.uk/174139/
From the Introduction: "However, the drivers of hatching failure are complex and poorly understood. In this review, we highlight the key factors associated with high levels of hatching failure beyond the impacts of predation, damage, desertion and exploitation. We then explore the underlying reproductive problems linked to hatching failure and how these are influenced by ecological and behavioural factors. We argue that a lack of understanding of the mechanistic basis of hatching failure can lead to flawed conclusions about how and why it occurs, with important implications for our understanding of avian ecology and conservation."