As someone who started with the D&DAS games and moved to D&D, here are some thoughts.
The D&DAS board games are not RPGs. They are board games inspired by D&D 4e, with the latest installment, Temple of Elemental Evil, also having a few basic elements from 5e. As you have noticed, if you expect an RPG experience from them, you’ll be disappointed.
They are, however, a great gateway to D&D. They share the basic mechanic of rolling a d20, adding a modifier, and comparing the result to a target number. Additionally, they are a fantastic source of minis to use in any edition of D&D. Specifically, Temple of Elemental Evil contains hero minis matching the classes and races of the pregenerated characters that are found in the 5e starter set.
Which brings us to the 5e starter set. This box includes a rule book containing the most minimal set of rules possible for 5e. It does not contain any rules for using a grid and minis. However, the accompanying adventure works perfectly well with a grid and minis. Basic rules for grids are found in the 5e Players Handbook or SRD, with more complete rules in the Dungeon Master’s Guide.
In order to utilize minis in the starter set, you will need to either draw or print (or build, if you’re really ambitious) each location where fights take place to scale (1″:5′). Most locations described in the book are accompanied by gridded maps which can either be drawn by hand to scale, or either scanned or purchased online for printing to scale. Those locations that don’t have maps in the book are easily drawn (a road with bushes to either side, a simple cave, etc).
Having transitioned from D&DAS games to the 5e starter set, I highly recommend giving it a try, adding in the rules for grid and minis from the PH and DMG. I also recommend following the starter set with Princes of the Apocalypse, a great adventure for 5e, which will utilize just about every mini you got in Temple of Elemental Evil, as they were released together.
However, coming from a war gaming background, 5e may not be your best choice. D&D 4e required much more tactical play than 5e, which is more optionally tactics-heavy. 5e’s more tactical play options (flanking, facing, more exact grid usage) are found in the DMG. D&D of any edition is, in the end, a pen and paper game. If your groups are not interested in this, but actually want a single unit war game, D&D is not right for you.
In summary, my advice to you is to take the 5e starter set, throw away the rule book it comes with, get the PH or SRD and the DMG, and give it a try, printing/drawing out the adventure’s maps to scale.