coughing widens the bronchi and leads to bronchiectasis ?

Infection/inflammation damages the airways.
Impaired mucociliary clearance leads to mucus pooling.
Structural damage causes airway dilation (pockets in airways, I liken it to diverticulitis but in the lungs)
Chronic infection perpetuates the cycle of inflammation and damage.

some snippets from keyword search (still no answer , what causes the dilatation)

Merck manual says : Bronchiectasis is an irreversible widening (dilation) of portions
of the breathing tubes or airways (bronchi) resulting from damage to the airway wall.
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Bronchiectasis is dilation and destruction of larger bronchi caused by
chronic infection and inflammation
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A balance between matrix metalloproteinases (MMPs) and tissue inhibitors of
metalloproteinases (TIMPs) may control the remodeling of extracellular matrix,
and excess MMPs have been associated with destruction or dilatation of airways
in patients with bronchiectasis.
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sputum levels of MMP-9 or TIMP-1 and their molar ratio.
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Cellular inflammation or possible imbalance between MMP-9 and TIMP-1 was not
demonstrated in this study to be related to bronchial dilatation in asthma.
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Irreversible bronchial dilatation on computed tomography (CT) is seen in patients
with interstitial pulmonary fibrosis whereas reversible dilatation of the bronchi
is a feature of many conditions.
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airways dilatation, which is a frequent finding in adult respiratory distress syndrome (ARDS),
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239 patients with bronchiectasis
In contrast to the traditional concept of "Bronchiectasis", our results suggested that
some adult bronchiectasis can be improved and completely radiologically reversed.
https://onlinelibrary.wiley.com/doi/abs/10.1111/1440-1681.13503
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The three most important mechanisms that contribute to the pathogenesis
of bronchiectasis are infection, airway obstruction and peribronchial fibrosis.
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Reversible bronchial dilatation is commonly encountered in viral and bacterial pneumonia
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In man, bronchiectasis predominantly affects the lower lung lobes, especially on the left side.
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In BALT lymphoma, the bronchial wall is not destroyed and the dilation is reversible

Coughing from infection and inflamation may lead to bronchiectasis, BUT coughing due to bronchiectasis also eventually causes tracheal collapse, called tracheal malacia. I have had bronchiectasis since I was 16 and I am now 74. That is a lot of years of coughing and clearing my throat. That has now caused tracheal collapse, which as you miight surmise, is a dangerous situation.