I will tell you, based on personal experience, when you send a message that says "Thanks", there is potential for that to be really, really annoying.
Let me explain why it can be even more annoying than the hypothetical scenario that you pitched. (Then I will explain why I say "Send it", despite having just proved how annoying it is.)
I've worked on some help desks that use electric tracking "tickets" to keep track of work that needs to be done. We can use our "ticketing system" (computer software) to send E-Mail. When we do, if a customer replies, their tickets gets highlighted as something that we, the staff of the help desk, need to reply to.
We go through dozens or hundreds of tickets a week. Some tickets are short and simple, while others can be quite aggravating to both our staff and the customers. When there are multi-day aggravating tickets, I'm very happy for them to go into a "Closed" status so that they aren't ever seen again...
until a couple of days later. As I'm struggling to not be behind schedule on some other important work that has cropped up, I notice that the ticket board has re-opened this ticket because the customer has replied. Seeing the subject of the highlighted ticket that I thought had been tossed down the bottomless pit forever, I notice feelings of dread creeping over me and overwhelming my very soul. The ticket was marked as closed and left our visibility, but apparently the customer had more to say. I think, "I thought we were done with this nightmare." Fearing the new details that may cause a new round of pain to surface, I tremble as I see just what additional pain has been conjured from the abyss's depths. Chills dance down and up my spine as I somehow I feebly manage to click on that ticket, and scroll to the area to see just what horrible addition the customer has added. And then I see it.
"Thanks! I appreciate the good work."
What? You're telling me that I have a bunch of really important work to do, but I interrupted my progress on that new important work, setting it aside to address an update to what was an even more important issue, and my workflow and progress has been utterly destroyed for a total of 8 seconds because... why? Because the customer had the audacity to thank me!
I have judged such replies to be pretty useless, really. In fact, harmful. Seriously, Heather and Jim and I all strive to keep our ticket board as empty as possible, and so if Jim gets a compliment, I'm not going to call his cell phone and interrupt his current progress on something else just to say "Alice said 'Thanks'." Nope, I just close the ticket, thereby cleaning up our board, and so Jim never even gets to see the compliment.
So, in the spirit of good customer service, I thought about how it would be good to train our customers NOT to send such "useless" replies. After all, if they slow us down, then they impact our ability to give better service to other customers. Since what comes around goes around, the practice eventually ends up being detrimental even to the customers that perform such atrocities.
However, when thinking about this, I realized just how horrible that would be. Basically, telling people, "don't be polite to us". We do want our customers happy with us, and so if they are spending time thinking good thoughts about us, it really is in our best interest to let the customers do this, despite the negative impact. (Finding a ticket, opening it, scrolling to the update location, and reading the text ends up taking me more time than just looking at my cell phone screen.)
So, even if your text does end up annoying the person by taking some of their time, sensible people should eventually see that your intent was to be nice, and this world could sure benefit by more people doing things with nice intent. Keeping a strong, good relationship developed is more important than some slight feeling of annoyance that a person may feel. So, even when "Thanks" may be least appreciated (e.g., frequently in Western culture), such an act is still worth doing. So, I conclude that yes, you should do this nice thing of sending a polite message. In this case, obeying Nike's slogan is recommended.