Welcome to the world of "at will" mercenaries.
I was once offered a job with a start date two weeks in the future, and then I got a call the day before I was set to start which told me I ought not bother showing up the next day, because the position had been cancelled. The job was with American Family Insurance, and instead of starting that new job, I instead called my agent to cancel all my policies with them.
So now, I have no reservations whatsoever about accepting a [non-contract] offer and making an appointment to show up for a new job, then continuing to interview right up to my new start date, or even a little while after. If I cancel the day before, the company should simply be glad that I didn't work one day then quit, because then they would have to process payroll and tax forms for the whole year.
Scheduling the first day costs me nothing. That's just a deadline for the company to put their written promise of payment for my labor into my hands. If it doesn't show up, neither do I. The company would then receive a call, and I would say that I was unable to start work because I didn't have a signed offer letter. If I get a better offer in the interim, I might accept it, and let the first company know they were outbid.
So far, I have only ever skipped a "start work" appointment with Deluxe, and I decided not to show up for work because their recruiter lied to me multiple times, and processed me in to a job description radically different from the one I had been seeking.
If you want assurance that I will show up after accepting an offer, give me an actual contract. An "at will" offer letter just says that if I work for you, you will pay me $X, N times per year.
Do not fear interviewing while employed. Do not fear it if you are "at will". Do not fear it if you are under contract. The company is not going to sacrifice a single penny of its profits for your personal welfare, so you have to see to your own financial security. You always have to assume that the contract will not be renewed or extended, or that the next all hands meeting will be to fire everyone. Therefore, you owe it to yourself to seek alternate employment at any time, and accept or reject offers in the spirit in which they were given. An "at will" offer could be for any number of days from zero until your retirement target, and that level of uncertain commitment from your employer is not sufficient by itself to generate any level of certain commitment from you.
You don't have to put bait on your hook, but your line should always be in the water. And if you catch a fish, you can always throw it back rather than eat it.