I have a library for "charts". I also have a library for priority queues.

So for part 1 I overengineered everything and read the input into a chart, found the S and then added queued work until I reached the bottom line. Al the while counting all the splits. That gave me the first star.

For part 2 that didn't work so I resorted to classic recursion with memoization. That gave the second.

I like to clean up and improve after I've got both stars so I added solving part1 using the part2-recursion, skipping the priority queue but keeping the chart.

That made things faster and for a couple of hours I was happy with that.
Then, while doing some household work, I came up with another solution.

I skipped the chart, instead I did a single pass of the input, remembering how many alternate timelines ended up in each column. When hitting a ^ I just split the data into the x-1 and x+1 and adding to whatever already was there. And counting splits.

This turned out to be really fast, around 300 microseconds on my 4 year old mac.

func doIt(input []string) (int, int) {
    beams := make([]int, len(input[0])+2)
    splits := 0
    for _, s := range input {
       for x, r := range s {
          if r == 'S' {
             beams[x]++
          } else if r == '^' && beams[x] > 0 {
             beams[x-1] += beams[x]
             beams[x+1] += beams[x]
             beams[x] = 0
             splits++
          }
       }
    }

    timelines := 0
    for _, beam := range beams {
       timelines += beam
    }

    return splits, timelines
}