If my US passport is stolen, will copy of first page verify passport number for travel, hotels, etc

If you have an iPhone, you can download a duplicate to wallet. It is not the same as a passport, but it has everything and identifier that your passport has.

A copy might work for some things, but hotels want an actual passport at check in. A copy won’t work for travel. If your passport gets lost or stolen you will need to interrupt your trip to go get a new one.

Isn't it possible to email yourself a copy of the passport pages?

This is what you need to do if you lose your passport while abroad: https://travel.state.gov/en/international-travel/help-abroad/lost-stolen-passport.html

I have never been asked by any hotel in Spain to provide my passport number ahead of check-in.
Has a hotel asked you for this in advance of arrival?

BTW: I almost always leave my passport in my hotel room, hidden in my suitcase or elsewhere.....I walk around with it only if I know that I need to show it at an attraction and in recent memory I have not had to do this, not even at Sagrada Familia in Barcelona a few months ago, or at the Palace of Catalan Music, same trip. You need it if you are doing heavy-duty shopping, though--for the VAT refund paperwork. And if you are taking a long-distance train (maybe for that..can't remember).

Every venue asking for identification will have their own requirements.

For example, The Alhambra website:

Each person is required to keep their ticket throughout the visit, and must present themselves along with an ORIGINAL personal identification document issued by the Ministry of the Interior or equivalent body of each country at the request of the staff...

For the Sagrada:

To gain entry, you must present your national ID card or passport to verify your identity. If you are entitled to any discount, you must also provide the relevant documentation to prove it.

Actually, Yes I have been asked by at least two hotels to provide my passport number before checking in. One hotel provided a WhatsApp phone number and the other used their presumably secure website. None of this answers the question I was asking, although it was somewhat reassuring to read some of the replies. If a passport was lost or stolen, of course, I would contact the US consulate and get another passport before I left Spain; my question was a bit narrower: will a copy suffice while I am in Spain?
Looks like a police report of the theft is also needed.
-Bruce

will a copy suffice while I am in Spain?

I don’t think anyone can answer this for you. With a police report and a copy of the stolen passport, the venue/hotel might show you some grace given the circumstances but obviously it would be at their discretion.

None of this answers the question I was asking

Well, Bruce, I believe my earlier post answered your question in regards to the Alhambra and the Sagrada Familia.

Whenever a hotel sends an email saying I can check in to save time, I ignore it. Those would require a passport number. Checking in doesn’t take very long.

I have never been asked by any hotel in Spain to provide my passport number ahead of check-in. Has a hotel asked you for this in advance of arrival?

I am nearing the end of 3 weeks in Spain. I have been asked to provide the physical passport everywhere we have stayed.

If advance check in is an option you might be able to just provide the number, but on more than one occasion I have also been asked to show the actual passport on arrival at the desk.

If they want a passport to prove your identification, they want the actual passport. No different than here in the US - law enforcement, TSA, etc. are not going to accept a photocopy in a situation that requires proof of identity.

To address your concerns about passport safety, there are many resources on this site for how to protect your valuables. Start here https://www.ricksteves.com/travel-tips/theft-scams