I would like to post a simple solution for addition of functions, probably understandable to most of mma users. Evaluate[ ] would calculate the result.

In[1]:= cs[x_] := Evaluate[Cos[x] + Sin[x]]

In[2]:= cs[x]

Out[2]= Cos[x] + Sin[x]

The method needs more work for combining functions that contain derivatives.

In[1]:= funcCombine[funcs_List, op_] := 
 Function[x, Evaluate[#[x] & /@ op @@ funcs]]

In[9]:= d1 = funcCombine[{D[#, x] &, D[#, y] &}, Plus]

Out[9]= Function[x$, 0]

I can combine total derivatives

In[10]:= d2 = funcCombine[{Dt[#, x] &, Dt[#, y] &}, Plus]

Out[10]= Function[x$, Dt[x$, x] + Dt[x$, y]]

In[11]:= d2[x y]

Out[11]= x + y + y Dt[x, y] + x Dt[y, x]

but that leaves me stuck with Dt, not D.

That seems promising. I wrote a function based on your idea:

In[20]:= funcAdd[funcs_List] :=
 Block[{temp, head},
  temp = head[x, #[x] & /@ Plus @@ funcs];
  Function @@ temp]

In[22]:= csf = funcAdd[{Cos, Sin}]

Out[22]= Function[x, Cos[x] + Sin[x]]

In[23]:= csf[1]

Out[23]= Cos[1] + Sin[1]

In[25]:= t = funcAdd[{#.# &, #.#/Sqrt[#.#] &}]

Out[25]= Function[x, Sqrt[x.x] + x.x]

In[26]:= t[Range[4]]

Out[26]= 30 + Sqrt[30]