Yes, MoPac dropped service to Galveston sometime between 1954 and 1955. The old GH&H has a decent alignment, but as you correctly note there are a plethora of grade crossings. Plus, the ABS was pulled out sometime in the late 1990s.
The GH&H line actually hosted a passenger service, called the Texas Limited, in the first half of the 1990s. Five (or so) restored lightweight cars and an F7, with T&E crew supplied by Amtrak (who was also operating a Houston section of the Texas Eagle during that time frame). It had some modest success in its first few years of operation, but Union Pacific refused to consider maintaining the tracks to any standard above Class 2. A two-and-a-half hour trip to Galveston didn't have a whole lot of appeal when you can drive that distance on the Gulf Freeway in little over an hour, even in weekend rush traffic. In addition, Amtrak wanted so much rent to sublease part of that pitiful small-town bus depot known as the Houston Amtrak Station that you would have thought it appropriate for a major tenant at Grand Central Terminal. About 1993 or so the Texas Limited operators moved their Houston terminal to a former Katy freight depot in The Heights a few miles northwest of downtown...but that made the trip times even longer.
I still wonder what might have happened had this operation shifted to the Santa Fe alignment through Alvin, but to the best of my understanding this was never seriously considered as the major investors in this service were from League City and wanted an en route stop there. Unfortunately, there's not a great deal of pent-up demand for 25 mph passenger train service between The Heights and League City....