Thanks, that's helpful.
I'd say it's more part of arriving at the lie, but it's all the same chunk of time.
You know where the disc is, so the lie will be behind it or behind a mini (if you mark it). You get a reasonable time to decide whether to mark it (which is what, a second and a half?) And a reasonable time to decide whether 803.02 applies.* Including checking with the rest of the card, I'd say.
And you get a reasonable time to get to that lie. A reasonable time to figure out how to get a foot into a bush so as to cause the least movement is more than what would be a reasonable time for stepping up to the disc in the open.
However, you don't get to mess around forever just because you haven't yet gotten to your lie.
(I always picture a five-year-old hovering their foot behind the marker saying "I'm not at my lie, I'm not at my lie, you can't start the clock yet!")
So, there could be a case where a player was taking sooooo long to get into that bush that the 30 second clock should have started.
*803.02 Relief from Obstacles
B. If an obstacle physically prevents the player from taking a legal stance behind the marker disc, or from marking a disc above or below the playing surface, the player may mark a new lie on the line of play immediately behind that obstacle.