Doctor Who was written for 26 years, almost entirely, by freelancers. The 21st century series' are also written mostly by freelancers, despite being in the Era of the Showrunner. So clearly, the way to get a line on what's been happening in the last 60 years is to follow each writer or writing team. This watch order is based on starting with the first serial, then following its writer's contribution to the show, then going back to the second serial, following its writer forward, and repeating. This is a frivolous idea that nobody should follow but I am curious how it would go all the same:
Anthony Coburn: An Unearthly Child (1963)
Terry Nation: The Daleks (1963), The Keys of Marinus (1964), The Dalek Invasion of Earth (1964), The Chase (1965), Mission to the Unknown/The Daleks' Masterplan (episodes 1-5, 7)(1965), Planet of the Daleks (1973), Death to the Daleks (1974), Genesis of the Daleks (1975), The Android Invasion (1975), Destiny of the Daleks (1979)
David Whitaker: The Edge of Destruction (1963), The Rescue (1965), The Crusade (1965), The Power of the Daleks (1966), The Evil of the Daleks (1967), The Enemy of the World (1967), The Wheel in Space (1968), The Ambassadors of Death (1970)
John Lucarotti: Marco Polo (1964), The Aztecs (1964), The Massacre (1966)
Peter R. Newman: The Sensorites (1964)
Dennis Spooner: The Reign of Terror (1964), The Romans (1965), The Time Meddler (1965), The Daleks' Masterplan (episodes 6, 8-12)
Louis Marks: Planet of Giants (1964), Day of the Daleks (1972), Planet of Evil (1975), The Masque of Mandragora (1976)
Bill Strutton: The Web Planet (1965)
Glyn Jones: The Space Museum (1965)
William Jones: Galaxy 4 (1965)
Donald Cotton: The Myth Makers (1965), The Gunfighters (1965)
Paul Erickson/Lesley Scott: The Ark (1966)
Brian Hayles: The Celestial Toymaker (1966), The Smugglers (1966), The Ice Warriors (1967), The Seeds of Death (1969), The Curse of Peladon (1972), The Monster of Peladon (1974)
Ian Stuart Black: The Savages (1966), The War Machines (1966), The Macra Terror (1967)
Kit Pedler/Gerry Davis/Elwyn Jones: The Tenth Planet (1966), The Highlanders (1966), The Moonbase (1967), The Tomb of the Cybermen (1967), Revenge of the Cybermen (1975)
Geoffrey Orme: The Underwater Menace (1967)
Malcolm Hulke: The Faceless Ones (1967), Doctor Who and the Silurians (1970), Colony in Space (1971), The Sea Devils (1972), Frontier in Space (1973), Invasion of the Dinosaurs (1974)
Mervyn Haisman/Henry Lincoln: The Abominable Snowmen (1967), The Web of Fear (1968), The Dominators (1968)
Victor Pemberton: Fury from the Deep (1968)
Peter Ling: The Mind Robber (1968)
Derrick Sherwin: The Invasion (1968)
Robert Holmes: The Krotons (1969), The Space Pirates (1969), Spearhead from Space (1970), Terror of the Autons (1971), Carnival of Monsters (1973), The Time Warrior (1973), The Ark in Space (1975), The Deadly Assassin (1976), The Talons of Weng-Chiang (1977), The Sun Makers (1977), The Ribos Operation (1978), The Power of Kroll (1978), The Caves of Androzani (1984), The Two Doctors (1985), The Mysterious Planet (1986), The Ultimate Foe (episode 1 only)(1986)
Terrance Dicks: The War Games (1969), Robot (1975), The Brain of Morbius (1976), Horror of Fang Rock (1977), State of Decay (1980), The Five Doctors (1983)
Don Houghton: Inferno (1970), The Mind of Evil (1971)
Bob Baker/Dave Martin: The Claws of Axos (1971), The Mutants (1972), The Three Doctors (1973), The Sontaran Experiment (1975), The Hand of Fear (1976), The Invisible Enemy (1977), Underworld (1978), The Armageddon Factor (1979), Nightmare of Eden (1979)
Robert Sloman/Barry Letts: The Daemons (1971), The Time Monster (1972), The Green Death (1973), Planet of the Spiders (1974)
Robert Banks Stewart: Terror of the Zygons (1975), The Seeds of Doom (1976)
Chris Boucher: The Face of Evil (1977), The Robots of Death (1977), Image of the Fendahl (1977)
Graham Williams/Anthony Read: The Invasion of Time (1978), The Horns of Nimon (1979)
Douglas Adams: The Pirate Planet (1978), City of Death (1979), Shada (Unreleased)
David Fisher: The Stones of Blood (1978), The Androids of Tara (1978), The Creature from the Pit (1979), The Leisure Hive (1980)
John Flanagan/Andrew McColloch: Meglos (1980)
Andrew Smith: Full Circle (1980)
Stephen Gallagher: Warriors' Gate (1981), Terminus (1983)
Johnny Byrne: The Keeper of Traken (1981), Arc of Infinity (1983), Warriors of the Deep (1984)
Christopher H Bidmead: Logopolis (1981), Castrovalva (1981), Frontios (1984)
Terence Dudley: Four to Doomsday (1982), Black Orchid (1982), The King's Demons (1983)
Christopher Bailey: Kinda (1982), Snakedance (1983)
Eric Saward: The Visitation (1982), Earthshock (1982), Resurrection of the Daleks (1984), Revelation of the Daleks (1985)
Peter Grimwade: Time-Flight (1982), Mawdryn Undead (1983), Planet of Fire (1984)
Barbara Clegg: Enlightenment (1983)
Eric Pringle: The Awakening (1984)
Anthony Steven: The Twin Dilemma (1984)
Paula Moore: Attack of the Cybermen (1985)
Philip Martin: Vengeance on Varos (1985), Mindwarp (1986)
Pip Baker/Jane Baker: The Mark of the Rani (1985), Terror of the Vervoids (1986), The Ultimate Foe (1986, episode 2 only), Time and the Rani (1987)
Glen McCoy: Timelash (1985)
Stephen Wyatt: Paradise Towers (1987), The Greatest Show in the Galaxy (1988)
Malcolm Kohll: Delta and the Bannermen (1987)
Ian Briggs: Dragonfire (1987), The Curse of Fenric (1989)
Ben Aaranovitch: Remembrance of the Daleks (1988), Battlefield (1989)
Graeme Curry: The Happiness Patrol (1988)
Kevin Clarke: Silver Nemesis (1988)
Marc Platt: Ghost Light (1989)
Rona Munroe: Survival (1989)
Matthew Jacobs: Doctor Who (TV movie, 1996)
Obviously I stopped with the 20th century stories, but the method is clear enough. When to bundle people who worked together and when to split them, as well as which stories to put where among those split, were done with my gut and no consistent rule.
Just putting this list together, one thing you immediately see is how writers love to bring back elements of their first or second stories, usually whatever alien. As freelancers they owned any original elements to their scripts, and had the right to sell them elsewhere if there was a buyer. Terry Nation tried to go international with the Daleks, but the Bob Baker and Dave Martin estates have made some cash off of K9. Also, the clear fact that when the show regularly did historicals, that was a specialty. There were certain people for the historicals and certain people for the sci-fi nonsense.
If one were to watch this way, I think you would get a solid handle on the most significant writers' takes on the material and their narrative fingerprints. Your Holmes, Hulke, Dicks, and the Bakers. But would siloing them all make it hard to see their influence on the writers to follow, especially those who did three or fewer stories and have less of a corpus to draw conclusions from?
This order probably erases the influence of the Script Editor and Producer on the show, but you see it implicitly. Anywhere there's suddenly a bunch of new names within a year or two, those jobs have changed hands and they're contracting new talent. Then you have very stable times like with nearly no new blood. There were as many first-time writers on the show in the last three seasons as in the entire 1970s.