Urgot Could Use Critical Strike & Attacks Speed Stats with Purge
When building Critical Strike & Attack Speed on Urgot 
, his Basic Attacks while using the Ability Purge deal less Damage per second in contrast to his non-Purge Basic Attacks. This appears offset when compared to an Urgot without Critical Strike & Attack Speed using Purge, which does cause him to deal more Damage per second.
Here are calculations for Urgot's Damage per second with and without Purge using Items that have Critical Strike & Attack Speed:
Urgot Level: 18
Purge Ability Level: 5
Applicable Runes: Eyeball Collection (full stacks), Transcendence (with Crest of Insight), Gathering Storm (3 stacks at 30 minutes), +10% CDR Offense Shard, +9 Adaptive Force Flex Shard (5.4 Attack Damage)
Items: Essence Reaver 
, Infinity Edge 
, Death's Dance 
, Trinity Force 
, Statikk Shiv 
, Elixir of Wrath 
, Rapid Firecannon 
Total Attack Damage: 511.2 = 131 + 18 + 48 + 28.8 + 5.4 + 65 + 80 + 80 + 25 + 30 Critical Strike Damage Multiplier without Purge: 1.25 = (100% + 25%) ÷ 100 Critical Strike Chance Multiplier without Purge: 2 = (25% + 25% + 25% + 25%) ÷ 100 + 1 Total Attack Speed without Purge: 1.71125 = 0.625 + 0.625 × (0.638 + 0.4 + 0.4 + 0.3) Damage per second with Basic Attacks (without Echoing Flames/Energized effects) without Purge: 2,186.9775 = 511.2 × 1.25 × 2 × 1.71125
Critical Strike Damage Multiplier with Purge: N/A Critical Strike Chance Multiplier with Purge (fixed): 0 Total Attack Speed with Purge (fixed): 3 Damage per second with Basic Attacks (without Echoing Flames/Energized effects) with Purge: 588.096 = (12 + 511.2 × 0.36) × 3
Non-Purge to Purge Damage per second comparison: 2186.9775 > 588.096 Percentage Damage per second increase from Purge Basic Attacks to non-Purge Basic Attacks: ~271.87% = 2186.9775 ÷ 588.096 × 100 - 100
Here are calculations for Urgot's Damage per second with and without Purge using Summoner's Rift Recommended Offensive Items that do not have Critical Strike & Attack Speed:
Urgot Level: 18
Purge Ability Level: 5
Applicable Runes: Eyeball Collection (full stacks), Transcendence (with Crest of Insight), Gathering Storm (3 stacks at 30 minutes), +10% CDR Offense Shard, +9 Adaptive Force Flex Shard (5.4 Attack Damage)
Items: The Black Cleaver 
, Ninja Tabi 
, Frozen Mallet 
, Death's Dance , Maw of Malmortius 
(without Lifeline), Mortal Reminder 
, Elixir of Wrath
Total Attack Damage: 494.2 = 131 + 18 + 36 + 28.8 + 5.4 + 40 + 30 + 80 + 50 + 45 + 30 Critical Strike Damage Multiplier without Purge: N/A Critical Strike Chance Multiplier without Purge: 0% Total Attack Speed without Purge: 1.02375 = 0.625 + 0.625 × 0.638 Damage per second with Basic Attacks (without Echoing Flames) without Purge: 505.93725 = 494.2 × 1.02375
Total Attack Speed with Purge (fixed): 3 Damage per second with Basic Attacks (without Echoing Flames) with Purge: 569.736 = (12 + 494.2 × 0.36) × 3
Purge to non-Purge Damage per second comparison: 569.736 > 505.93725 Percentage Damage per second increase from Non-Purge Basic Attacks to Purge Basic Attacks: ~12.61% = 569.736 ÷ 505.93725 × 100 - 100
As the math concludes, building Critical Strike & Attack Speed causes Urgot's Basic Attacks without Purge to deal more damage than with Purge, almost 275% more; not building Critical Strike & Attack Speed causes his Basic Attacks to deal approximately 12.61% more damage with Purge. This makes Purge ineffective while building Critical Strikes and Attack Speed, which can be the staples of a high Physical Damage dealing Champion that uses Basic Attacks as a main source of damage. In my opinion, building Critical Strike & Attack Speed and still being able to viably use Purge should be an option. To do this with balance, Basic Attacks enhanced by Purge could deal more Damage based on Urgot's non-Purge Total Attack Speed if it is greater than 1 (which would still be affected by Attack Speed buffs/debuffs based on non-Purge Total Attack Speed, but with a minimum threshold of 1 and a cap of 2.5 unless uncapped by other sources such as Lethal Tempo or Hail of Blades) and be able to Critically Strike in the same way non-Purge Basic Attacks can.
The formula for Purge using Critical Strike & Attack Speed would look like this:
Purge Damage per second at Ability Level 1/2/3/4/5 = ((Purge Base Damage + Total Attack Damage × 0.20/0.24/0.28/0.32/0.36) × Critical Strike Damage Multiplier٭ × Critical Strike Chance Multiplier × Non-Purge Total Attack Speed٭٭) × Purge Attack Speed
٭ If Critical Strike Chance Multiplier is > 1 ٭٭ If non-Purge Total Attack Speed is > 1
Here is an example using the previous Critical Strike & Attack Speed build:
Total Attack Damage: 511.2 = 131 + 18 + 48 + 28.8 + 5.4 + 65 + 80 + 80 + 25 + 30 Critical Strike Damage Multiplier: 1.25 = (100% + 25%) ÷ 100 Critical Strike Chance Multiplier: 2 = (25% + 25% + 25% + 25%) ÷ 100 + 1 Non-Purge Total Attack Speed: 1.71125 = 0.625 + 0.625 × (0.638 + 0.4 + 0.4 + 0.3) Purge Damage per second (without Echoing Flames/Energized effects): 2,515.9482 = ((12 + 511.2 × 0.36) × 1.25 × 2 × 1.71125) × 3
Purge to non-Purge Damage per second comparison: 2,515.9482 > 2186.9775 Percentage Damage per second increase from Non-Purge Basic Attacks to Purge Basic Attacks: ~15.04% = 2,515.9482 ÷ 2186.9775 × 100 - 100
Here is an example without Critical Strike & Attack Speed using the previous Summoner's Rift's Recommended Offensive Items build:
Purge Damage per second (without Echoing Flames): 583.26723 = ((12 + 494.2 × 0.36) × 1.02375) × 3
Purge to non-Purge Damage per second comparison: 583.26723 > 505.93725 Percentage Damage per second increase from Non-Purge Basic Attacks to Purge Basic Attacks: ~15.28% = 583.26723 ÷ 505.93725 × 100 - 100
I'm not exactly sure how much balance or lack thereof these exact formulas would bring to the game, but I generally think that implementing Critical Strike & Attack Speed into Urgot's Purge is a worthwhile investment. Both the Attack Speed & Critical Strike portions would cause Purge to be useful against non-structures and the Attack Speed portion would cause Purge to be useful against structures, which translates these stats from non-Purge efficiencies to Purge efficiencies. Keep in mind that although these changes would "buff" Urgot, Purge would still have the drawback of less Base Movement Speed. The main purpose of this change would be to make Critical Strikes & Attack Speed viable with Purge, because Purge is an Ability that changes how Urgot's Basic Attacks function and currently renders him at a severe loss (less Base Movement Speed and ~275% less Damage per second) if used with Critical Strike & Attack Speed.
