I suspect with a small firm that doesn't necessarily have a dedicated travel "guy" it probably simplifies things letting people book their own flights.
For some people, airline choice is a big deal. For some cities, you might even pick the wrong airport (i. e. if the person has an easier commute to Newark or Laguardia than JFK). Not to mention scheduling (maybe someone wants to travel the "day before" because of risk of too-tight scheduling).
OTOH, maybe it's simpler for "local team all in one city" versus "remote workers being herded to a single event".
For one thing even if the company never reimbursed you - it wasn’t an outright scam, you did get a flight. The scam scenario you are literally just sending money to a scammer - with the extra flourish of getting a fake check.
What you say about expensing is not unusual and there a number of reasons for it. It does offload the onus of correctly making reservations, putting in KTIN and immigration info on you. And then the big one is that I get all the travel card benefits (points). I even prefer it this way.
This is no more inherently scammish than my relationship with my credit card company or any time one agrees to pay later (which is the norm of how businesses with supply chains operate).
The point is there’s no need to use some ill defined concept of weird. Where is the money going and what is the purpose? If someone is giving you an unbacked check and requesting a wire transfer to themselves it’s a scam.
While agreeing to be paid later for an unrelated good or service has some amount of risk it isn’t a scam.
And there are clear benefits to booking your own plane tickets.
a) you’re buying tickets for someone else
b) that scam company also owns the airline (or if you’re buying from a third part online agency)
Otherwise how could you get scammed? Haha we tricked you into going to Atlanta but didn’t monetarily benefit from it at all!
For example, when I company I worked for sent me on a business trip, I was supposed to buy the flight ticket using my own money, then bring them then receipt, and then they would pay me the money back. Why couldn't they buy the flight ticket themselves instead? They knew when and where I am traveling; they were the ones deciding that. It would be easier for one HR person to buy 5 tickets at once whenever one of the teams went to a business trip, than for 5 developers to buy the tickets individually and figure out the rules.
This totally felt like a scam (use your own money to buy something we want to buy, and you only have our word that will get the money later back), except it didn't make sense, because it was a large international company employing thousands of people, and I have already worked there for months. And yes, it was perfectly legit. Just needlessly complicated and weird.
So the fact that a company does something weird and needlessly complicated doesn't necessarily make it a scam. It's just likely a scam when they do it before paying you the first salary.