Formalities for an American spouse of an EU national staying in France for 4 months annually

Yes, it is necessary. On a formal level, there is no gap in the law, more than 90 days is a long séjour for which third-country nationals need either a visa long-séjour valant titre de séjour (VLS-TS) or carte de séjour (or a carte de résident but that one is definitely not relevant here). As impractical as it is with all the delays and staffing problems in the bureaucracy, there is no intermediate status between residence and short stay.

I think the VLS-TS is mostly intended for people who cannot enter France without visa and especially for people who do not benefit from the EU freedom of movement (e.g. third-country nationals who are the spouse of a French citizen), which would rule out your mother. You could however still ask the relevant French consulate about it.

If your mother could get one, the requirements would be pretty much the same as for the carte de séjour, but the advantage is that it could be arranged in advance from the US each year, sparing her the generally unpleasant experience of dealing with the service des étrangers of the préfecture during her summer break. Otherwise, the only solution is the carte de séjour.

Either way, a stay longer than three months is a long stay, with all associated consequences. But note that there is no unified concept of "residence" under French law. It's judged independently for immigration, tax or citizenship purposes. Thus it's perfectly possible to be deemed a resident for tax purposes and hold carte de séjour for many years and yet see an application for naturalization fail because you do not meet the (more stringent) residency requirement for this purpose.

I don't know what the rules are for driving licences but to the extent that it is necessary to exchange them after some period of time, your mother's immigration status would therefore not necessarily make a difference. I think the embassy might be oversimplifying this by tying the requirement to exchange the licence to the carte de séjour.

Obviously, none of this solves the six-month waiting time issue but I don't think there is anything in place to make it easier for people in this situation. The only thing I can think of is that if your parents have a summer residence, it's perhaps in a rural département where things might be better than in, say, Seine-Saint-Denis. In practice, six months is merely a target that is frequently violated but it can also be quicker I think. (Beyond the six-month rule in EU-law, there is a 4-month implicit refusal for all carte de séjour that is routinely ignored).

Also, I don't see anything stopping your mother from applying for a carte de séjour and then leaving. The next year, she can either pick it up and immediately file for an extension (if the card is still in the préfecture, in spite of the directive) or perhaps start a new application. As crazy as it sounds, as long as she always has an application pending before going over the three-month threshold, she would not be staying illegally.

Researching another question, I came across article R222-3 of the code de la route, which seems to be the basis for the need to exchange a driving licence:

Tout permis de conduire national, en cours de validité, délivré par un Etat ni membre de la Communauté européenne, ni partie à l'accord sur l'Espace économique européen, peut être reconnu en France jusqu'à l'expiration d'un délai d'un an après l'acquisition de la résidence normale de son titulaire. […] Au terme de ce délai, ce permis n'est plus reconnu et son titulaire perd tout droit de conduire un véhicule pour la conduite duquel le permis de conduire est exigé.

To the extent that I understand all this correctly, it means that the licence only loses its validity one year after one establishes one's “normal residence” in France. And, in this context, normal residence is defined in article R222-1 as

On entend par "résidence normale" le lieu où une personne demeure habituellement, c'est-à-dire pendant au moins 185 jours par année civile, en raison d'attaches personnelles ou d'attaches professionnelles.

Your mother's situation (only 4 months a year, no specific professional or family ties with France) clearly does not meet this definition, even if she requires and does hold a residence permit. It's very different from the notion of residence used in the law on entry and stay (where three months is the relevant threshold) or taxes (where the definition is much broader). So the driving licence might be a non-issue.