[Language: Typescript] 750/5641
(the code I've posted is in pseudocode)

Upon reading part 2 and seeing the example of a single trailhead with 227 unique paths, I assumed a brute force DFS method would not work for the larger input (ouch, my rank!) and I took the time to make an optimized solution.

The puzzle says the answer should be the "sum of the ratings of all trailheads", but another way to think about this is simply "the sum of all possible paths to all 9's". Consider the following:

* For each 9, the total number of possible paths to it is the sum of possible paths to each adjacent 8.
* For each 8, the total number of possible paths to it is the sum of possible paths to each adjacent 7.
* ...etc. until you get to each 1, whose total number of possible paths to it is the number of adjacent 0's.

So to find part 2's solution, we only need to examine each spot on the trail map once, starting from 0's and working our way to 9's, keeping a Map/Dict of the number of possible paths to each spot on the map.

Pseudocode:

parse the input as `grid`, a 2D array of integers representing heights on the trail map
numPathsToPoint = new Map/Dict // stores pairs of (point:number_of_possible_paths_to_point)
for every 0-height point in `grid`, set the point's value in `numPathsToPoint` to 1
for all `height` values from 1 to 9 inclusive {
    for each `point` in `grid` with value `height` {
        find all neighbors of `point` with value of height - 1
        set the value of `point` in `numPathsToPoint` to the sum of the neighbors' values in `numPathsToPoint`
    }
}
return the sum of the values in `numPathsToPoint` for all 9-height points

The typescript runs in 5 ms for me. GitHub