It still is a bird in hand sort of thing. If you like the first place that you visit, how tempted do you become to look at other offers?
If we are one of the first 2-3 teams that have the player in town, the opportunity is the greater selling point. If it is clear that Arizona is a place that you could find yourself in as a player, don't be the 6th or 7th team that gets the visit, because it is very likely that the player's decision is known in the first 2 or 3 visits.
Mind you, I have actually been in the process of changing jobs since the start of 2019. I looked up some positions elsewhere, dusted the resume off, and began applying heavily around mid-January. Didn't feel like I was benefiting in my position at all any more. I left my job before having a firm offer in hand, and recently started a new job. In the end, I had a few companies that I had to say, "Thanks, but I have to cancel this upcoming interview as a result." I guess that my logic is that there may have been nothing wrong with these other companies, and maybe they were capable of offering more services, money, benefits, etc. But because they weren't first to the table, they would never get that opportunity.