For Ship P&E I use WS&IM rules
For Overland Tracking P&E I use OS rules, the pursuit scenario card (or search if not tracking)
For Normal Encounter P&E (where creatures can sense each other and on foot or riding) I use my own personal version cobbled together from early D&D.
AD&D's P&E rules are far from perfect, but if you read DMG pages 67-69 you'll see it has some particular sensibilities. The chase scene wasn't the reason for running IMO. It really was more about getting away or stopping someone from doing so. There is still an opportunity for Round by Round actions for changing the odds on evading or catching another, but a good bit of one's action economy (so to speak) is usually taken up in sprinting.
FWIW, if you add in the fatigue rules for running as a strenuous activity, then this stuff gets is often quickly settled with dice rolls. However, having the opportunity do resolve it via ingenuity some other way, round-by-round before chance ultimately decides is what makes this version enjoyably playable rather than "They get away".
"They get away" & "Roll to grab 'em" do happen often enough anyways given the disparity of movement speed (and maneuverability and other stuff) between monsters and at at least the bog standard PC races. Sometimes you're just faster, sometimes they are. Sometimes the dragon can just fly away, sometimes the wizard teleports.
