ApexPC:Putting counts for a sizable % of your score, so upgrade your putter first.
While your arguments are true for experienced players, it may be different for beginners.
Putting includes two elements: The putter and the putter. The Starter putter is one of the best clubs in the set, not so much worse than a low level custom putter. I see no need to rush with the equipment while the player lacks the experience to bring the putts close, not to speak about holing out difficult ones.
OTOH, the Starter wedges are of the worst clubs in that set. A beginner who has problems to ding and to get the shots to the green will face a lot of short game and will be lost with those clubs. Their forgiveness is nil and their imprecision is legendary. They are very nonlinear when choked.
IMHO, the quickest way to get a decent score is by discarding those §$%&% wedges. Custom wedges respond to the shots in a fair way and provide some forgiveness. They are by far better from rough or green bunkers.