I'll suggest going the Arduino way and relying on magnetic contact switches - fairly inexpensive, and invisible once embedded in a build.
Another commonly used trick is using an ultrasound distance sensor on the receptacle - the distance will be (near) zero if there's anything attached.
Are there any standard bricks with contacts built-in[?]
Yes - the 9V system which started in the late 80s featured 2x2 plates with conductive metal inserts, e.g. part 5306bc011 "Electric, Wire with Brick 2 x 2 x 2/3 Pair, 11 Studs Long" (besides battery packs, train motors and lights):
- Each wire is connected to both two studs on the top and two antistuds on the bottom, and
- 9V-compatible parts are not being manufactured anymore, and therefore are only available in the aftermarket
- Any 9V parts in the aftermarket are at least a decade old, and cables are subject to wear&tear.
Since you are not afraid of using glue, and if you're keen on going the "closing an electric circuit" route, you might instead want to use copper foil tape (which should be readily available at crafts stores), or even conductive ink.
Finally, if you want an unconventional approach with contemporary available parts, you can always use Mario pants.