Hi!
I'm just a "regular user" and I just wanted to post my outside point of view on this situation:
AMR is a wonderful tool for aggregating manga reading across all manga sites. It not only allows you to have manga from different web sites but it also sync's your reading list on all your devices. From a user point of view this is almost perfect (still not on your phone or tablet

). So I guess that part is hardly debatable.
Now the problem is the other side of the coin: the manga providers - or the web sites that may get hurt by the usage of AMR.
There was previously stupid attritions between AMR and some manga providers like mangastream and the way it went was just wrong (basically MS tilled the mangas and putted them together with JS and according to the dev at that time they weren't very open to discussion of compromises).
What I purpose is a compromise: state the requirements for AMR mechanism (advertising banners loading rules, image loading rules, what calls are necessary for your points system, usage of top (or left) banners when reading, etc). This way you allow your readers to use AMR and minimize/obliterate the negative impact it has on your website. It can also potentially improve your website performance and lower bandwidth usage since it will not require to load all theme resources for each page.
If people are using AMR on your website it's because it goes toward their needs and instead of blocking it try to welcome it

Everyone can win from this!