Pretty much yes, but that'll change when the 3rd America class LHA comes out with a dock making it an LHD.
Another way to think about it would be that it's the role of the LHD*/LHA/LPD to secure the beachhead then the LSD brings the majority of the rest of the equipment onto the beach.
*LHD has vehicle carrying capacity too including tanks, AAVs, artillery and the like.
If this PDF i'm looking at is accurate (I can't find a date and some of the vehicles have been upgraded since), then the ground element of an MEU consists of approximately
- 4 M1 Abrams tanks + M88 recovery vehicle
- 15 AAVs
- 8 - 17 M1161 Growlers
- 31 MTVs (7T trucks)
- 7 - 17 LAV-25s (+ 1 EW LAV)
- 105 HMMMWVs
- Unknown number of towed artillery guns
- Load of logisitics vehicles, trucks, bulldozers and the like
That's a lot of vehicles to move, so a prime vehicle mover with a dock like an LSD is ideal. In the UK LSDs are built cheaper and manned by members of the Royal Fleet Auxiliary so they provide cheaper sealift than our LPDs.