I already asked a similar question here but unfortunately didn’t get much feedback with suggestions to make a better use of that time. In my case it was when I was doing an exchange with a friend (Hindi for Japanese) and I felt just like you, that there is a lot of waste of time. I think a more ideal situation would be if that time was actually a time to practice the language, with conversation instead of explanation but that wasn’t what usually happened, maybe similar to your situation.
He was a beginner in Japanese and I was a beginner in Hindi, so instead of having time to practice, we needed to give a kind of class to each other, with different degrees of success. I have all of that material recorded on video, but like you said, I kind of feel it like a waste of time to go over it again, because it’s not much material anyway, and things that I could actually learn by myself or get much more input through listening/reading intensively.
The interaction with a native speaker is of course great, but I would probably enjoy it better if I already had at least a B1 level.
What gave me more input was that when he started doing a kind of immersion method through stories, pictures etc. and then asking questions at the end, " a la LingQ", so we could decrease the English speaking time and explanations, except when really necessary.
Another problem I faced was with a Chinese friend, that had a much higher level of Portuguese (my mother tongue) than my poor Chinese. She had graduated in Portuguese at University and I had only HSK 3 and she didn’t have any teaching experience or didactics, what probably you are not facing by paying a teacher at iTalki, though you can find teachers with different levels of skills there.
I wouldn’t care much about checking and reviewing the words for each lesson, I prefer the teacher to increase the input through immersion methods without the need to memorize/comprehend 100% of it. If the teacher doesn’t know how to do it, maybe you’d better choose another skilled one (if that is your preferred method) or suggest a natural approach method that could increase your input without much explanation in English. You could also set a “homework” time, where you could pre-study about a certain topic, text, video, etc. and then you would talk about that, go through a story together, etc. The teacher could give you a summary of the main expressions, new words in a pdf/doc for your reference, which is faster to look up than going through a recorded lesson again. Or the text, if you used that as a base for conversation.
Though some people don’t like to be corrected, I like the teacher to point my mistakes and correct my pronunciation, I think that’s the main reason for having a teacher, to be corrected and be able to clear your doubts…the words you can always check by yourself (unless the tutor is teaching you slang and informal language that you won’t easily find elsewhere in a dictionary or reference books)