American English: Must I always use a singular pronoun with a collective noun?

Your usage is correct. It is rare in the US (and Canada, to my knowledge) to refer to grammatically singular nouns in the plural, whether with generic or proper nouns. (Congress is debating, Google has released X, etc.)

The only common exception I can think of is "audience", where the verb will still agree with a singular subject "the audience is watching", but the pronoun "they" can refer to the individuals of the audience. This is especially common with the possessive pronoun, e.g., "The audience watched with awe, biting their nails as the act went on".

This is in part due to the unnatural sound of "it" when the object of the verb (nails) references the humanity of the subject (audience). This also relates to the use of singular "they", whereby we get "everyone in the audience was biting their nails" or even "the audience was biting their nails". "Their" acts as a gender-neutral pronoun. When "everyone" is used as the subject, using "they" is correct as the gender-neutral pronoun, where other authors may prefer "he" or "she". (The APA recommends against this since the 7th edition of the manual.) When "everyone" is not included and "audience" is the subject, the use of "they" does, indeed, refer to the group noun, bringing us back to this common exception to US aversion to plural reference to singular group nouns. (See parliament, group, team, family, and Congress for other examples.)

Keep in mind, though, that while this usage ("audience" + "they") is casually accepted, it will not fly in formal US English. (This is in reference to using plural verbs and pronouns with grammatically singular nouns. The use of singular "they" has gained academic acceptance and is the only singular third-person pronoun recommended by the APA when the gender of the person referred to is unknown or irrelevant.)

Americans are perfectly happy to use they and their as pronouns for collective nouns. What they do not do is use plural verbs with collective nouns. We would tend to say:

My family was fighting among themselves,

and not

My family were fighting among themselves,

or

My family was fighting among itself.

even though in the first formulation, family changes from singular to plural halfway through the sentence.

I don't have any formal grammar book stating this, but you can find some evidence for this from Google Ngrams here and here. When you set the corpus to American English, Google Ngrams finds hits for was fighting among themselves, but none for was fighting among itself, and some for the team shares their, but none for the team shares its.

Of course, you can also refer to the family or team with the pronoun its, and if you can consider the family or team as a singular entity in the sentence, that's generally what is done. But your sentences treat the team as a group of individuals, so I would tend to use the pronoun they.