That said, it doesn't really matter what anyone else does, right? This is between you and the company. Some companies are we'll-reimburse-you shops, some will want to book your ticket, largely because they have internal processes built up to do one or the other. (A note on market norms for young HNers: you should expect to get plane tickets and a hotel room for any job interview in a city which requires flying, courtesy of the company. A company not offering that is a company to avoid. I personally feel like companies owe candidates a per diem -- modest payment to defray the cost of meals and incidentals -- but you'll find that relatively few companies spontaneously offer this.)
Most companies should be fairly reasonable with regards to making alternate arrangements if you're not in a position to do them yourself. You do not have to elaborate on why. "Hey $RECRUITER, Thanks for the email. It would be more convenient for me if you could book the tickets/hotel room. What information from me do you need to make this happen?" will probably work. You will almost certainly not be the only prospective employee to have ever made that request in history.
If one does not have a pressing reason why one would do this, I would suggest one just let companies run their standard processes. It makes things modestly easier for you. Also, if you have e.g. space available on credit cards, "I pay and then invoice you actual costs" is strictly superior to "You arrange travel" -- you get reward points (whee) and, more importantly, reasonable discretion with regards to choice of flight and accommodations.