The configuration you describe is a common one for gas plants. I think it is common to have only a loop seal to drain liquid out of the primary absorber and prevent vapor back flow. If vapors were to be entrained with the liquid from the primary absorber, they would recycle back to the primary absorber from the high pressure separator. This recycle could increase the vapor load to the primary absorber, if it were excessive.
I do not believe there are any process safety related hazards with loosing the level in the primary absorber. There might be a concern for two-phase flow in that rich oil line so it might vibrate if there was a significant vapor flow through that line.