What can go wrong if we let sigma algebra to admit the union of uncountable union of elements? By definition we only allow the union of countable infinite of elements to be also include the $\sigma$ field, why not uncountable many? Is there a historical view behind this？

If we require sigma algebras to be closed under arbitrary unions, then any sigma algebra on $\mathbb{R}$ that contained the open intervals would necessarily contain every subset of $\mathbb{R}$.
In particular, things like Vitali sets would have to be measurable.
As a consequence, any translation-invariant measure on $\mathbb{R}$ would have to either (a) be the zero measure or (b) assign infinite measure to finite intervals.
This would make measure theory kind of useless.
