"He will gladly compare them."
Translation:Er wird sie gerne vergleichen.
The penny drops.
If you think of the sentence as He will + gladly compare them then it comes out as er wird + sie gerne vergleichen. .
Think of the verb as gladly compare and the auxiliary as he will, rather than he will gladly + compare.
Yes, glücklich is state of being, Du bist glücklich, etc.
When used as an adverb, happily, gladly, etc. You have to use gern, or gerne.