You are talking about agent nouns.
VERB STEM+er does that for many German verbs, technically. But it does not always make sense or sound natural:
läuten → Läuter?
regnen → Regner?
zerschlagen → Zerschläger? or Zerschlager? (some people will refuse the second version because "Schlager" already exists as word for "popular song")
Sometimes, it describes (predominantly) something for doing something:
bohren → Bohrer (drill)
schlagen → Schläger (bat, racket, stick)
mixen → Mixer (kitchen mixer, less frequently used in context with beverages or music)
Furthermore, the transformation does not always describe someone who
schauen → Schauer (to look and shower)
füllen → Füller (to fill and filler/pen)
bauen → Bauer (to build and peasant)
lenken → Lenker (to steer or drive and handlebar; "Lenker" actually does have "driver" as second meaning, but mainly in Austria and Switzerland)
Here are some guidelines the community has come up with so far:
Despite the two counterexamples above, the verb to be transformed should actually describe some acting.
bleiben, meinen → Bleiber?, Meiner?
If the verb ends on
-eln the Suffix is
sammeln, basteln, wickeln → Sammler, Bastler, Wickler
Pay attention on verbs ending on '-rn'. Their stem is everything but the 'n'. zaubern → Zauberer
For a female agens,
-in is appended.
Bastler, Mörder, Werfer → Bastlerin, Mörderin, Werferin
A in the stem often becomes ä if the verb has strong inflection.
tragen, backen, verraten → Träger, Bäcker, Verräter
but: beraten → Berater (even though beraten is conjugated the same way as verraten)
On the other hand, sometimes also weak verbs change a to ä.
jagen → Jäger
In its present form, this "rule" cannot be applied to au.
tauchen (weak), rauben (weak) → Taucher, Räuber
Although Räuber could be an exception like Sänger.
One example for o becoming ö was found, yet its inflection is weak.
morden → Mörder
U seems not to change, even with strong verbs: rufen → Rufer
Note that singer was normal German, too (and Meistersinger is still known to contemporary speakers thanks to the popular opera by Wagner). It was replaced by Sänger. So singen does not actually become Sänger, it's just that the two words are so old, they've undergone different developements.
Another similar but not equal possibility is to use the first participle of the verb as a noun:
zerschlagen → der Zerschlagende
Although you dismissed this possibility, I think there are some cases where it works as you want it to:
der Wissende (one who knows)
der Suchende (one who searches)