Travelling to the US as the spouse of a citizen

For peace of mind I was looking up too and found this on website of the US embassy to Austria:

FAMILY-BASED EXEMPTIONS

Certain travelers are exempt from COVID travel restrictions including:

• Legally married spouses of U.S. Citizens (USC);
• Legally married spouses of Lawful Permanent Residents (LPRs, also known as “Green Card” holders);
• Under-21, unmarried child of a USC or LPR;
• Parent or guardian of an under-21, unmarried USC or LPR;
• Under-21 unmarried sibling of an under-21 unmarried USC or LPR.

-Unfortunately, unmarried partners, “Partnerschaft,” fiancées, fiancés, adult children, parents, grandparents and extended family do not qualify for family exemptions.

Travelers who qualify for Family Exemptions do not need NIEs or any notification to this Embassy. You may proceed directly to the airport and bring supporting documentation that proves your relationship to the U.S. Citizen or LPR relative, such as marriage certificates pr birth certificates.

IMPORTANT: DO NOT CHECK INTO YOUR FLIGHT ONLINE! Doing so may inadvertently cancel your ESTA. Arrive at the airport at least three hours before your flight with all your required documentation.

Admission to the United States remains subject to a determination by Customs and Border Protection (CBP) at the U.S. Port of Entry. A valid visa, ESTA or NIE is not a guarantee of admission.

Here's a quick attempt at addressing some of the points raised in your question. But first,

Be very careful trying to use ESTA to fly to the US from a restricted country, including the Schengen area.

In short, if you do not make prior arrangements to establish that you are the spouse of a US citizen, your ESTA will probably be cancelled when you try to use it to check in for your flight.

See While the covid-19 regulations are in effect, can I enter the US if I am married to a US citizen and have an ESTA? (This user asked a question similar to yours, and, three days later, edited the question to add "My ESTA got canceled!")

Also related: Travelling to USA (NYC) from UK to see my daughter - what documentation is required?

The FAQ page of the US embassy in Spain does not discuss cases directly related to yours. There is information about applying for a "national interest exception," but that exception is established by the proclamation's paragraph 2(a)(xii), whereas the spouse exception is established by 2(a)(iii). Unlike the national interest exception, the spouse exception is not discretionary, and it should not require an application.

If I were you, however, I would write to the e-mail address given on the embassy's web site. Although an application should not be necessary, it does appear to be necessary to register one's status as the spouse of a US citizen and to state one's intention to travel to the US in order to prevent the ESTA from being cancelled. From the FAQ page:

I believe Presidential Proclamation 10143 does not apply to me / I believe I qualify for an exception. What should I do?

If you have a valid visa or ESTA and believe, after reading the proclamation and the updated information in the FAQs, that you qualify for an exception, please send an email to [email protected]

Due to the large volume of inquiries, we will only respond to those who have imminent travel and who qualify for an exception. Please note travel for the primary purpose of tourism remains suspended.

(emphasis in original)

A similar message from the US embassy in Luxembourg supports this conclusion:

Any traveler with a valid ESTA who attempts to travel to the United States in violation of this Proclamation will have their ESTA canceled. ... Individuals [who] believe they are exempt from the Proclamation and are abroad may contact their air carrier for additional guidance. Travelers who believe they fall within an identified exemption are encouraged to seek guidance in advance of planned travel to avoid travel disruptions.

(emphasis added)

The page of the embassy in Spain repeatedly says "tourist travel remains suspended," but this doesn't actually follow from the executive order. You might have to wrangle with them a bit. You might try saying that you are traveling for medical treatment, not tourism, but since that's the same visa category as tourism it might not help. Legally, though, the proclamation exempts spouses of US citizens regardless of their purpose of travel, so it shouldn't matter what the purpose of your trip is.

Now, on to your question:

do I have to travel with my wife?

There is nothing in the proclamation that requires you to travel with your wife, nor her to be in the US. However, consular officials might think that this is a requirement, just as they might think that spouses of US citizens are only allowed to enter for certain purposes, so try not to mention your wife's location if you can avoid it, but be prepared to argue. If they try to assert either restriction, ask them to name the paragraph of the EO containing that restriction.

I know CBP has broad discretion and can deny entry for any reason

That's not actually true, but it might as well be, because if a border officer decides not to admit you for an improper reason (such as the color of your shoes, or perhaps something more pernicious), the officer can instead claim that the denial of entry was based on a permissible reason, and there's no real way to challenge that -- especially if you're seeking admission under the visa waiver program (that is, having traveled to the US with ESTA authorization).

This might all be more trouble than it's worth, but if you do attempt it, please come back and post an answer describing your experience.