Can I be rated for a time when my rater was not even on the book?

Your last NCOER ended 31 MAY 2020, so your current NCOER must start on 1 JUN 2020 whether 1 JUN was rated time or not. There are no gaps in NCOER dates, from and to dates cover every single day of NCO time, rated or not rated.
If your rater want to be pedantic, or is just uncomfortable about rating your performance in JUN-JUL-AUG prior to his arrival, he can put those dates down as norated time code Q, but since he more than meets the minimum rater time, he can cover the entire annual IAW AR 623-3.
Also, any time during the year you were at a 1059 producing school would be nonrated time code S,

Per 623-3, USAR raters only need to be the supervisor for 120 days. As far as the timeline goes, it could go both ways. No the person cannot rate you if they are not in the unit. However, if you refuse to have an NCOER from that person since they weren't there for a month in the rating period, you could potentially change your NCOER from an annual to a change of rater -- this may be easier said than done if the person that was actually in charge of you has already left.

Also, for what its worth, I personally would not want a change of rater NCOER with just 1 month of production.

You want to make sure you do not have any unrated period of time in your NCOER history.

Evals seem to be hit and miss in the USAR depending on unit S1's reminding CoC's to knock them out, and this last year of COVID has made them a mess. I'm rating NCO's I have never personally met face to face for a year of Battle Assemblies that were all done virtually.

The best Evaluations you are going to get are the ones you participate in writing yourself. That could mean spoon feeding your rater every single bullet comment for each section, to simply providing a list of your accomplishments, or completely depending on them to do the whole thing from scratch (I don't recommend this).

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Now, the way I read your post, it sounded like your rater was a 2LT. So they probably have never done one before. Don't know the process, don't know the system, and don't know the left right guidelines of each part of the evaluation. So even if the 2LT did their best attempt it's probably going to get kicked back and forth by HRC for various errors.

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Here's what I'm getting at. Have at the ready the NCOER draft that you would like to see in your evaluation history and iPerms.

Then, when your higher chain of command is getting the heat put on them for late Evals and a COL is forcing the issue and just assigning raters and Sr raters to your DOD# you can make your rater's job easy and CAC signatures can be applied, and your eval can be sent up to HRC for review.

Then once it is iPerm'd all is one with the world and the gears keep turning in the USAR.

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This is the window of opportunity many soldiers don't take advantage of in the USAR. Too many blow it off, when they can have an otherwise strong Eval verses some generic bullets a promotion board will clearly identify as minimal performance of a non flagged soldier.