quantum field theory - Why does local gauge invariance suggest renormalizability?

I'm reading Gauge Field Theories: An Introduction with Applications by Mike Guidry and this particular remark is not obvious to me:

A tempting avenue is suggested by the QED paradigm, for if a local gauge invariance could be imposed on the weak interaction phenomenology we might expect the resulting theory to be renormalizable. [Guidry, section §6.5, p. 232]

Is there an obvious argument for this "local gauge invariance suggests renormalizability" remark? I should add that I still tend to get lost in the streets of renormalization when unsupervised, i.e. I'm not familiar enough with the entire concept to have any real intuition about it. (references on renormalizability that might help are of course also welcome)