Here is my report for primogem and fate income in June 2021 in order to help you better plan out your pulls!
A few key points:
YMMV: "Your-Millage-May-Vary", gems in this category would still technically count as F2P, but depending on the player, it could either be hard to obtain (like Spiral Abyss or limited-quantity promo codes,) or yet to be obtained due to differences in progress (like time-gated/gacha-related content, hidden achievements, world exploration.) Gems in this category are based on my personal progress.
Archipelago world quests and the quest-related chests are not counted in YMMV, since it should be easy to get them all by looking up guides. The rest of the archipelago chests are counted as YMMV.
Gems from teapot companion gifts in this month depend on how many characters you have obtained. I've decided to calculate based on 24 obtained characters (excluding Traveler,) which I think is a good baseline for most players.
"Marine Hues" Battle Pass is based on level 40 completion. Level 50 completion is only possible in July without the purchase of Gnostic Chorus.
Genesis Crystals from Welkin are excluded since they can be used to buy other items like skins.
Pity calculations are using 82 soft pity (based on data from the Wish Tally project; most pities occur before 82.)
As usual, if you believe something is missing, or if you have any suggestions, please comment and let me know.
In case you're looking for past counts:
May 2021
April 2021
March 2021
February 2021
January 2021
November and December 2020 (courtesy of u/fffeeelll)

This Month's Summary:

Total F2P Event Banners:
33.53 to 51.03 wishes
avg. 1.12 to 1.70 wishes per day
5364 to 8164 primogems equivalen*
**% to 62% pity
Total Paid Event Banners (including F2P):
54.40 to 71.90 wishes
avg. 1.81 to 2.40 wishes per day
8704 to 11504 primogems equivalent
66% to 88% pity
Verdict: Another best month yet, second only to February. Islands exploration and Teapot Gift Set contributed lots.

F2P Primogems:
1800: Daily Commission
60 per day * 30
1610: Event
120: Mimi Tomo
60 per sighting * 2 (last 2 days are in June)
20: Klee Test Run
690: Midsummer Island Adventure
240: Event Quest
60: Part I - Mysterious Islands: Journey to the Unknown
60: Part II - Summer Vacation: Proceed With Caution
60: Part III - Dodo-King of the Sea: Lying in Wait
60: Part IV - The Final Riddle: A Secret Uncovered
90: Main Cannon, Make Ready... Fire!
30: Collect 200 Shiny Flotsam
30: Collect 1,000 Shiny Flotsam
30: Collect 2,000 Shiny Flotsam
180: Whirlpool off to Starboard... Full Speed Ahead!
30 per challenge * 6
90: Samurai Sighted... To Arms!
30: Defeat Maguu Kenki * 3 times
90: Harpastum Bombs Loaded... Blow 'Em Away!
30 per 3 monster camps cleared * 3
360: Echoing Tales
60 per 4 Echoing Conches * 6 (up to 24 Echoing Conches)
360: Legend of the Vagabond Sword
60 per challenge * 6 (Day 7 is in July)
40: Distant Voyage (Web Event)
20: Kaedehara Kazuha Test Run
660: Compensation
300: Version 1.5 server maintenance
300: Version 1.5 bug fixes
60: 6/25 iOS login issues
389: Quests
309: World Quests
40: They Who Hear the Sea
40: The Winding Homeward Way (10 from chest)
69: Who Wields the Wild Wind? (19 from chests)
50: From Outer Lands (10 from chest)
40: The Other Side of Isle and Sea
70: A Trip Through Fog and Wind (30 from chests)
80: Chapter II: Prologue - Autumn Winds, Scarlet Leaves
60: Quest Reward
20: Achievements
5: The Gathering Storm (Earn the chance to go to Inazuma on board the Alcor.)
5: Ready, Fight! (Earn your first victory in The Crux Clash.)
10: Autumn Winds, Scarlet Leaves (Complete "Autumn Winds, Scarlet Leaves.")
60: HoYoLAB
60: Daily Check-In (20 * 3)
30: Teleport Waypoints
5 per Teleport Waypoint unlock * 6 (first point is not unlocked by player)
15: Guaranteed Achievements
5: Yo-Ho-Ho, and a Bottle of Dandelion Wine (Climb aboard your Skiff.)
5: Mighty and Illuminated Wind Rider (Continuously sail your Skiff for a certain period of time.)
5: Fast and Furious (Continuously sail your Skiff at high speeds for a certain period of time.)
Total F2P earning so far: 4564 (28.53 wishes, 35% pity)

F2P Your-Millage-May-Vary Primogems:
1200: Spiral Abyss
50 per 3 Abyssal Stars * 36 stars * 2 rounds
960: Serenitea Pot Companion Gifts
40 per Character, assuming 24 out of 32 obtained characters
485: Exploration (Non-quest Related)
110: 2 per Common Chest * 55 (excluding 2 which can be picked up during quest)
285: 5 per Exquisite Chest * 57 (excluding 1 which can be picked up during quest)
90: 10 per Precious/Luxurious Chest * 9 (excluding 6 which can be picked up during quests)
105: YMMV Achievements
10: Wonders of the World
5: Nice Boat! (Switch Skiffs with another player.)
5: ...And Her Name Is the Mary Celeste (Suffer the destruction of your Skiff...)
20: A Realm Beyond: Series I
20: Not Just a Small Bench (Create 600 furnishings.)
75: A Realm Beyond: Series II
5: Honored Guest of the Realm (Invite a companion to stay in your Serenitea Pot.)
5+10+20: Fireside Chats (Unlock 10/20/30 interactions with your companions)
5+10+20: Gifts All Around (Receive 5/10/20 gifts from your companions)
50: Promo Code (limited quantity)
Genshin Impact v1.6 50 Primogems Giveaway
Source 1: iGames (still available as of writing)
Source 2: SEA** (ended)
Source 3: SteelSeries Games (ended)
Source 4: Crucial Games (still available as of writing)
Source 5: Intel® Gaming Access (still available as of writing)
Total F2P earning including YMMV: 7364 (46.03 wishes, 56% pity)

Paid Primogems:
2700: Blessing of the Welkin Moon
90 per day * 30
Total paid earning:
including F2P: 7264 (45.40 wishes, 55% pity)
including F2P+YMMV: 10064 (62.90 wishes, 77% pity)

Fates:
9: Acquaint
4: "Marine Hues" Battle Pass
5: Stardust Exchange (YMMV)
11% pity, equivalent to 1440 primogems
5 or 9: Intertwined
0 or 4: "Realm of Diversion" Battle Pass (paid Gnostic Hymn required)
5: Stardust Exchange (YMMV)
6% or 11% pity, equivalent to 800 or 1440 primogems

Additional Personal Stats. for Reference (Welkin+BP)
Primogems:
25: older achievements
7: older exploration
26 characters for gift sets
Personal total: 10176 primogems, 72.60 event pulls (equivalent to 11616 primogems, 89% pity)


