If you're doing a real engine break-in the economics of it alone rule out synthetic oil, never mind the other reasons for not using it as a break-in oil. If you really want to go over the top and the cost isn't an issue, Gibbs oil sells oil specifically for engine break-in.
Shop By Product - Break-In & Assembly Oils - https://drivenracingoil.com/c-1389195-shop-by-product-break-in-assembly-oils.html So does Maxima:
Maxima Oils #30-10901 Premium Break-In - MaximaUsa.com - https://maximausa.com/series-263-premium-break-in.html
Most of the articles about engine break-in are talking about cam-in-block engines where the cam and follower's wear pattern is a poorly lubricated, highly loaded set of mating parts. OHC engines like the 22R/E don't have this design and it's cam and followers are not nearly as highly loaded. What they have to say about piston rings and cylinder bores does apply to all engines.
With the bores & rings, cam & followers having already established their wear patterns a full engine break-in process is a bit excessive, BUT since cleaning is the reason for opening it up I'd still run it for a short time period (long enough to get thoroughly warm) on dino to flush out all of the junk that you didn't get and then go to synthetic with a new filter. I would use a quality oil for the 'break-in" because you do want an excellent detergent package in the oil that you do use.
If you really want to go down this particular rabbit hole have a look at:
Bob Is The Oil Guy - https://bobistheoilguy.com/forums/