How do I make this loop all children recursively?

Please try this function. Seems to be do what you are looking for.

(inputtable as table, searchvalue as text, inputtext as text)=>
let 
input = searchvalue,
childrows = try Table.SelectRows(inputtable, each [Parent] = input)[Child] otherwise {},
output = if List.IsEmpty(childrows) then inputtext & Text.Combine(childrows, "") else Text.Combine(childrows, "") & Text.Combine(List.Transform(childrows, each @fnParentChild(inputtable, _, "")), "|")
in 
output

If needed, you can see it with your data, you can put this in a blank query.

let
    Source = Table.FromRows(Json.Document(Binary.Decompress(Binary.FromText("i45WclTSUXIC4oAgfxcQ08s5VClWJ1rJGchxQYi71HiChUHKvbAo98Kt3BmLcpB4KBZxEMcTIQ40JBYA", BinaryEncoding.Base64), Compression.Deflate)), let _t = ((type nullable text) meta [Serialized.Text = true]) in type table [Parent = _t, Child = _t, Param = _t, Children = _t]),
    #"Changed Type" = Table.TransformColumnTypes(Source,{{"Parent", type text}, {"Child", type text}, {"Param", type text}, {"Children", type text}}),
    #"Added Custom" = Table.AddColumn(#"Changed Type", "Custom", each let par = [Parent] in fnParentChild(#"Changed Type", par, ""))
in
    #"Added Custom"

Pat