It might be helpful to be more specific. There is nothing that can be described as the Local Government Act. Much of the legislation that affects local councils is in the Local Government Act 1972. But there are earlier and later acts that also affect local councils, and as you indicate some of the later acts amend the 1972 act.
The SLCC gives a firm and explicit answer to the questions at https://www.radyr.org.uk/council/wp-content/uploads/sites/3/2016/02/Co-opted-member-as-chairman.pdf
As I understand it, the legislation may not use the word "co-option" but has provisions that carry the import of "If no poll is claimed in time, the council fills the vacancy as soon as practicable". This process is widely known as co-option, as you say. The element "as soon as practicable" may also not be in the legislation, but lawyers will advise that if a local authority has an obligation to do something, then unless there is a good reason otherwise, what it is obliged to do should be done as soon as practicable. The law is not just the statutes, it is also common law, case law and lawyers interpretation of how statutes should be read (in effect, opinions on how a judge would read the statutes were a case to come to court).
I agree that co-option is a second rate way of appointing councillors, and should be avoided so far as possible. However, the cost of a poll runs to thousands of pounds, and could become a significant burden on citizens. Also, I think it would be difficult to sustain interest in repeated by-elections for parish councils.