Nope, not the case at all. A lot of people advise to only send a letter of intent if you already have an MD acceptance - I personally disagree with that because if that school is your #1 choice and you will absolutely go there even if you get another acceptance, I believe that’s important information they should know. That’s just my own opinion that can be considered objectively wrong *shrug*
But a letter of interest can be sent at any time. A lot of people will send them even if they haven’t gotten an interview yet to show they’re still interested. Some send them after interviews to demonstrate that they’re really fond of and serious about the school - without the commitment that comes with a letter of intent. Some people send them after they get waitlisted to hopefully increase their chances of getting accepted off of the waitlist compared to an applicant with similar stats who did not take extra measures to demonstrate interest.
Basically my take on it: this whole process is rather subjective and you should take any action you personally deem acceptable and worthwhile. Want to send a letter? Do it. It’s your future on the line so that’s your call to make - not others’