What is the role of "much" immediately after a noun?

Much isn't attached directly to the verb. It is just part of the adverbial phrase much of the time. You can replace this phrase with an adverb, such as always to get

Child development specialists believe that confining babies always to strollers, high chairs, playpens, and walkers can inhibit muscle development.

When you do this you discover what is causing the confusion: the word order has to be different for an adverb and for an adverbial phrase, so whilst this is the correct order with the phrase, my sentence is incorrect and you have to reorder it to

Child development specialists believe that always confining babies to strollers, high chairs, playpens, and walkers can inhibit muscle development.

As for the adverbial phrase itself, much is being used as a noun. This is quite a common sort of adverbial phrase with words that indicate proportion:

some of the time
all of the time
none of the time
most of the time
etc.

often with words that are usually adjectives such as some, all, most.

"Confining babies" is not a noun phrase. Gerund-participles are verbs, not nouns, and "confining" is a gerund-participle in that sentence. An entire clause headed by a gerund-participle acts "like" a noun phrase (NP) in certain respects (it can be used as the subject of a clause, as in your sentence, or it can be used as the object of a verb or preposition), but it is not considered to be an NP.

"Much of the time" has the same function here as it would in a sentence like "When parents confine babies much of the time to strollers, high chairs, playpens, and walkers, it can inhibit muscle development."

If I remember correctly, this kind of adverbial element is analyzed as modifying the verb phrase/VP (which in your sentence is headed by the verb "confining").