Replace sum x+y+z in expressions like 2x+3y+z

You can use PolynomialReduce for this:

reduce[e_, p_Symbol->r_, v_List] := Module[{min},
    min = First @ Ordering[Coefficient[e,#]&/@v];
    Replace[
        PolynomialReduce[e, r, v[[min]]],
        {{n_}, s_} :> s + n p
    ]
]

Then:

reduce[3x + 2y + 5z, p -> x + y + z, {x, y, z}]
reduce[3x + 2y, p -> x + y + z, {x, y, z}]

2 p + x + 3 z

3 x + 2 y

You can build another kind of rule

rule = a_. x + b_. y + c_. z :> (((a - min) x + (b - min) y + 
                               (c - min) z + min p) /. {min -> Min[a, b, c]})

Then

(5x+2y+3z /.rule) == 2p+3x+z
(3x+2y /.rule) == 3x + 2y

Clear["Global`*"]

repl[expr_, vars : _List : {z, y, x}, p : _Symbol : p] :=
 SortBy[expr /.
     (Solve[Total[vars] == p, #][[1]] & /@ vars) // Simplify,
   LeafCount][[1]]

5 x + 2 y + 3 z // repl

(* 2 p + 3 x + z *)

3 x + 2 y // repl

(* 3 x + 2 y *)