B757 Captain reported landing safely after experiencing multiple systems failures associated with the loss of hydraulic system 1.

Narrative

On ILSXXR; Gear Door EICAS go around initiated. Noted low Hydraulic Quantity left system. Conducted non normal IAW FM; consulted [Maintenance] and Dispatch. On second approach; had Auto Speed Brake failure. Pre-briefed possibility as a precaution; continued approach and landed. Emergency equipment was standing by.About an hour after departure; we noticed 0.76 on Hyd Sys 1. Pressure was good. I set my watch for a timer; about an hr later … 0.75RF. We discussed consequences of loss of Hyd Sys 1. On approach to ZZZ; we had a Gear Door EICAS after lowering gear. After a thought; descending thru 1;900 was contacted Tower and conducted a go around. We left the gear down and climbed to 2000 runway heading. Switched controls FO now flying; CA running managing non-normal. Briefed Flight Attendants; Dispatch; [Maintenance]; ATC and passengers. We discussed what could happen if System 1 failed; covering T/R; speed brakes; etc. We received vectors. Switched controls; now CA flying; FO PM. On landing checklist; the Auto Speed Brake EICAS illuminated. I reset it; the. It came back. We discussed that on landing; throttle idle; speed brake; then T/R … expecting T/R would fail (they did not). Below 1000; AT had though time managing speed. Aircraft was loose and felt wobbly; like aileron flutter. At the time; we did not consider the drag of the gear door being out. I choose to just fly the aircraft and disconnect auto throttle.