Most of the "thanks" comments that you are flagging will be instantly killed off by the system and never make it to the moderator dashboard.
There will be very little impact on human moderators based on your SEDE query criteria. You will see most of these flags will make the comment instantly vanish. The Stack Exchange Network has dedicated scripting to assist with the removal of "thanks" comments and the placeholder text on every Stack Exchange community explicitly says not to say "thanks" or "+1". This is because these comments are useless noise to researchers (these are the users that we are ultimately trying to give value to).
As for any comments that are not instantly deleted, my opinion is:
"Don't worry about the donkey (mule/horse), load the cart (wagon)."
If you see something that doesn't belong and/or doesn't benefit researchers, flag it for removal.
If there aren't enough moderators to handle the flag load, the existing moderators can ask for more moderators and/or better tools. That said, I don't think a minor spike in flags is going to greatly disrupt an entire team of moderators. This, after all, is just housekeeping that was waiting to be done.
If certain communities are retaining useless/noisy comments, then they are not only creating unnecessary page bloat. They are teaching their citizens that making noise is tolerated/acceptable. Then when noisy commentors spread their wings and join a new community, they will take their bad habits with them.
We all have a responsibility to groom each other for the betterment of the whole. By minding our patch and guiding users how to ask, answer, and comment, we put Stack Exchange on a positive trajectory. By better "training" users in our own communities, these users become helpful hands when they join new communities. This is a positive butterfly effect.
Finally, if we don't serve up heaps of flags, then how will we find our next @Bhargav (the world's most prolific flag smasher)?
I just just unearthed an old mJSE post of mine which was posted after I joined a new community and before becoming a moderator there. While it is a little long-winded, it demonstrates the frustration of an experienced Stack Exchange user joining a community and being confused/horrified at the moderator team rejecting absolutely valid comment flags en masse. My question goes pretty far to offer an itemized record of flags which should have been deemed helpful, but were not. My post also serves up nine reasons why we should be trashing noisy comments. Why don't JSE moderators trash unnecessary comments like other SE sites?