Cannot invoke initializer for type ‘Range’ with an argument list of type ‘(Range)’

Some background:

In Swift 3, additional range types were introduced, making a total
of four (see for example Ole Begemann: Ranges in Swift 3):

Range, ClosedRange, CountableRange, CountableClosedRange

With the implementation of SE-0143 Conditional conformances in Swift 4.2, the “countable” variants
are not separate types anymore, but (constrained) type aliases, for example

 public typealias CountableRange<Bound: Strideable> = Range<Bound>
      where Bound.Stride : SignedInteger

and, as a consequence, various conversions between the different
range types have been removed, such as the

init(_ other: Range<Range.Bound>)

initializer of struct Range. All theses changes are part of the
[stdlib][WIP] Eliminate (Closed)CountableRange using conditional conformance (#13342) commit.

So that is the reason why

let range: Range<Index> = Range<Index>(start..<self.endIndex)

does not compile anymore.

How to fix

As you already figured out, this can be simply fixed as

let range: Range<Index> = start..<self.endIndex

or just

let range = start..<self.endIndex

without the type annotation.

Another option is to use a one sided range
(introduced in Swift 4 with SE-0172 One-sided Ranges):

extension String {
    func index(of aString: String, startingFrom position: Int = 0) -> String.Index? {
        let start = index(startIndex, offsetBy: position)
        return self[start...].range(of: aString, options: .literal)?.lowerBound
    }
}

This works because the substring self[start...] shares its indices
with the originating string self.

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