Delta used to allow Diamonds to board with First Class, which could sometimes create massive hurds of people moving towards the boarding door all at once, but recently changed it so Diamonds board later in the process. So, if you are exclusively buying First Class, the boarding process is somewhat OK when it is time for First to board.
However, keep it mind First Class is not really the first to board:
Pre boarding / extra time is first. Sometimes this is an insane amount of people.
Military with ID is next, which again can be a huge group. Personally this irritates me because at certain stations this can literally be half the plane.
Sometimes I hear First Class called next, other times I hear families with car seats and strollers. Again this can be a real zoo.
At this point First is called if it hasn’t been before:
Next is Comfort+, Sky Priority, Main 1, Main 2, Main 3, Basic, and finally all remaining customers.
Even when I’m in First, and need overhead bin space, I do find myself being a bit more assertive to board, because even in F, the bin space can dry up. I usually try and check as much as possible to avoid this, but sometimes that’s not an option.
DL does, like every airline, still have an issue with people crowding the gate area during boarding. The other night is Seattle boarding in First on a redeye to ATL it was like playing Frogger just to get to the boarding door due to all the people standing around that presumably had Main 1, 2, or 3 boarding passes.
Oh, and lastly, I haven’t seen DL’s Sky Priority lane used during boarding in a long time. Seems to be just the general lane.