Why would a time signal receiving watch have an airplane mode?

Most useful radio receiver designs utilize some form of the superheterodyne architecture, where one or more radio-frequency local oscillator signals are used to shift the frequency of the received signal to an intermediate frequency where it is more readily processed.

Typically, the local oscillator signal will radiate from a receiver to some degree - how much depends on the specific design, but it is not uncommonly detectable in near proximity.

Restrictions on radio frequency sources on passenger airlines have historically tended to err on the side of caution, restricting anything that could theoretically emit moreso than being specifically tailored to demonstrated concerns. Since a local oscillator theoretically radiates, the argument was that you should not be operating one, which effectively meant that you should not be operating a radio receiver.

Of course digital processing circuitry typically radiates clocks and their harmonics, too... But then, the shift in the last several of years towards services like in-flight WiFi also indicates a shift in thinking away from vague theoretical concerns.

I think that there is a very simple answer to this question that has nothing to do with electronics. A number of airlines prohibit the use of GPS receivers in flight. Their rationale does not actually matter. In order to comply with their rules, the watch must be able to turn off the GPS receiver somehow or it would be prohibited from being on the plane entirely.

Here is a list of airlines which do not allow GPS receivers: http://gpsinformation.net/airgps/airgps.htm

Does GPS actually cause interference? Doubtful. http://gpsinformation.net/airgps/gpsrfi.htm

If you wish to keep GPS enabled during flight, check with the pilot.

Mobile phones, GPS receivers, laptops or indeed smart watches have no effect whatsoever on avionic equipment. Most kit is designed and tested for restricted emissions and EMI tolerance in reverse.

The reason this equipment is banned has nothing to do with safety, but financial expedience. There's a lot of electronic equipment in the world. To rigorously prove my assertion in a way that would absolve an airline in a damages court case would take £100Ms. No one wants to pay that, especially if it would have to be passed onto customers. Better just ban the stuff. People will still keep flying as they're a captive market. You'll fly naked sat in a transparent plane if necessary.

Try leaving your phone fully on the next time you fly a long way across Europe. If you survive to land at the other end, you'll find perhaps 10 -15 messages on it welcoming you to various mobile networks that you over flew. No problem.

Edit following comments:

There are approximately 1 million people flying in the air at any one time across the world. If 1 in 1000 people randomly forget to switch off their e devices in flight (plausible), this theory is tested 1000 times/day across all randomly selected airliners in the world. That's 365,000 times a year on every conceivable aircraft type. That's a million hours /year of testing based on a 3hr mean flight time (plausible). That's a pretty extensive systematic experiment. Nothing dangerous happens.