From a completely different perspective, there is Jean-Eric Pin's article [Tropical semirings](https://hal.archives-ouvertes.fr/hal-00113779/file/Tropical.pdf), and then
Stéphane Gaubert's short introduction [Methods and Applications of $(\operatorname{max},+)$ Linear Algebra, Report 3088, January 1997, INRIA](https://hal.inria.fr/inria-00073603/document) and a longer set of lecture notes by the same author, [Introduction aux Systèmes Dynamiques à Événements Discrets](http://amadeus.inria.fr/gaubert/PAPERS/POLY12-02-1999.pdf) (these are in French but are excellent lecture notes). The point of view of Gaubert is to describe _discrete event systems_ which are great fun and very approachable. Later when discussing the applications in that area some of the points that you made in the original question can be seen but with much greater emphasis on the applicability. See also Jean-Pierre Quadrat, _[Max–Plus Algebra and Applications to System Theory and Optimal Control](http://www.cmap.polytechnique.fr/~gaubert/PAPERS/ICM94-preprint.pdf)_, Proccedings of the International Congress of Mathematicians, Zurich 1994,  Birkhauser, 1995.

I learnt a lot from a talk by Jeremy Gunawardena many years ago, and for further information you could look at his _Idempotent Semi-rings_  and his website (which is easy to find).  

Another useful website is http://www.maxplus.org/ linking into several groups working on the (max,+)-algebra.

Note none of this really needs the algebraic-geometric side as such, at least to start with, and is much more linked to ‘systems theory’,  and applications such as combinatorics, scheduling, and dynamic programming problems in Operational Research.  It is also great fun both to learn and to teach.