Here's a solution that manipulates language rather than strings. You and others might also find op_literal() useful in the future.

Solution

Helper: op_literal()

This helper function op_literal() actually manipulates the R language itself to repetitively use a binary operator like + across many operands...even though a binary operator typically accepts only two operands. Calling op_literal(`+`, w, x, y, z) will actually generate this expression here: w + x + y + z.

# Helper function to arbitrarily repeat a binary operation (like '+').
op_literal <- function(op, ...) {
    # Capture the operator as a symbol.
    op_sym <- rlang::ensym(op)
    # Count the operands.
    n_dots <- rlang::dots_n(...)
    
    # Recursive case: a binary operator cannot handle this many arguments.
    if(n_dots > 2) {
        # Split off the final operand.
        dots <- rlang::exprs(...)
        dots_last <- dots[[n_dots]]
        dots <- dots[-n_dots]
        
        # Perform recursion for the remaining operands.
        op_left <- rlang::inject(op_literal(
            op = !!op_sym,
            ... = !!!dots
        ))
        
        # Assemble recursive results into the full operation.
        substitute(op(op_left, dots_last))
    }
    # Base case: the binary operator can handle 2(-) arguments.
    else {
        substitute(op(...))
    }
}

Note

Since op_literal() generates an expression, you still need to evaluate it if you want the result:

op_exp <- op_literal(`+`, 1, 2, 3, 4)
op_exp
#> 1 + 2 + 3 + 4

eval(op_exp)
#> [1] 10

Custom Function: print.f()

Next, this custom print.f() then leverages op_literal() to assemble the formula:

# Your custom 'print.f()' function.
print.f <- function(data, var1, ..., group) {
    # Capture the core variables as symbols.
    group_var <- rlang::ensym(group)
    other_vars <- rlang::ensym(var1)
    
    # Count the additional variables.
    n_dots <- rlang::dots_n(...)
    
    # Append those other variables if they exist.
    if(n_dots > 0) {
        other_vars <- rlang::inject(op_literal(op = `+`, !!other_vars, ...))
    }
    
    # Assemble the formula.
    formula_exp <- rlang::inject(~ !!other_vars | !!group_var)
    
    # Generate the table according to that formula.
    table1::table1(
        formula_exp,
        data = data
    )
}

Result

Given your dataset reproduced here

dataset <- data.frame(
    ID = c(1, 1, 2, 2, 3, 3, 4, 4),
    group = c("gp1", "gp2", "gp1", "gp2", "gp1", "gp2", "gp1", "gp2"),
    col1 = c(0, 1, 1, 1, 0, 1, 1, 0),
    col2 = c(0, 0, 1, 1, 1, 1, 0, 0),
    col3 = c(1, 0, 1, 0, 1, 1, 1, 0)
)

your call to print.f()

print.f(dataset, col1, col2, col3, group = group)

should yield the following visualization:

Note

As it stands, you have defined the group parameter at the end of your function header. This means that if you try calling print.f() like so

print.f(data = dataset, var = col1, col2, col3, group)

then you will get an error: without the group = specification, that final variable gets lumped together with col2 and col3, all under the ... umbrella. This will generate a bad formula:

~ col1 + col2 + col3 + group | 

To avoid the pain of having to type out group = every time, you can simply relocate it before the ..., like so:

print.f <- function(data, group, var1, ...) {
  #                       ^^^^^

Once you've done so, the following call will work as you intended:

print.f(dataset, group, col1, col2, col3)