From what I have found, each party member will gain EVs even if they did not participate in battle,
If you don't want to be constantly updating your party, this leads two effective strategies to be:
- Farming money to buy Vitamins.
Vitamins are items that increase EVs all the way up to the max amount.
But they do cost quite a bit, so you will need to farm up some money in game to buy them all.
- Sending Pokemon on seminars from the Job Board in Pokemon Centers.
This allows you to train up to 10 Pokemon per stat at once, but it takes real world time to do.
More detailed information can be found in my source, but I have also now tried all of this myself to confirm.
So the most effective way mostly depends on your play-style:
With lots of play time you can just farm up money with an Amulet Coin (found behind the Motostoke outskirts sign) and buy all the vitamins,
you can also play the Battle Tower and buy vitamins with BP.
Vitamins purchased can instantly fully EV train a Pokemon, you would need 25 for your two stats, bringing them to 250 each, then either a 26th vitamin or one hour with the below method to hit the 252 max.
Or you can log in daily and send up to 10 Pokemon off on a seminar, then return the next day and do it again. From my experience sending a Pokemon off for a whole day rewards 96 EVs. There are however mulitple time options to suit your playstyle, all awarding 4 EVs per hour:
Timeframe Text Actual Time EVs Obtained With PowerItem
Whole Day 24 hours 96 288
Half Day 12 hours 48 144
Very Long 8 hours 32 96
Long 4 hours 16 48
Short 3 hours 12 36
Very Short 2 hours 8 24
Just a little 1 hours 4 12
As noted on Serebii, having your Pokemon hold one of the Power Items will also increase the EVs obtained from the jobs.
Buying these power items is a great investment, each adds +8 EVs per hour to the Pokemon holding it, effectively tripling the amount of EVs gained per hour!
It will not take much game time, but this method could be more real world time as it will take several days to fully train a Pokemon, unless you have the power items, then it can take just over two days (one full 24 hours for each of the two stats, then you use another hour for the remaining third stat for 4 EVs).
Your calculations seem correct to me; based on serebii, with a 25+ combo and at least 500 of that species battled, you have a 33% chance to get the higher shiny rate. So overall you have a (1/3)X(7/4096)+ (2/3)X(1/4096) chance, about one in 1365. (I'm not entirely sure how the rerolls are worked out - if it simply rolls the 1/4096 chance seven times looking for at least one shiny, the odds are instead one in 1366).
Assuming you were at full odds the whole time, then, the chance of 2000 encounters with no shiny is 23.1%. Not all that unlikely. Once you get up to 6289 encounters you'd be looking at just under 1% chance of still no shiny (still fairly possible).
Best Answer
The Shiny Charm returns once again this Generation and increases the regular encounter shiny rate from 1 in 4096 to 1 in 1365.3. It can be obtained by completing the Pokedex and then talking to a doctor within a room on the 2nd floor of Hotel Ionia in Circhester. The Masuda method has also returned with no significant changes as far as we know. Chaining however has been replaced by a new system:
The more times you've defeated a certain species of Pokemon in battle, the higher your chances are of finding a shiny of that species, with the best odds being at 500+ battles. These battles do not have to be in a row (chained) to receive these increased odds. You can keep track of how many times you've battled a certain species of Pokemon from their Pokedex entries as it has a counter. For example, at 500+ battles won against one species, your chances of getting a shiny are increased to 1 in 683 (1 in 512 with the Shiny Charm). The graph at the bottom of this page should help. *
*Serebii, which has been a reliable source of Pokemon information for years, is claiming it's possible there's a bug in the code causing the new shiny hunting method not to be anywhere near as effective as it should be. In the table linked you linked there is a "Chance of Shiny Rate" column. This column shows that at 500+ encounters you only have a 3% chance of receiving the boosted 1 in 683 odds. Serebii has also confirmed that's what the column means on their official Reddit account. Just some anecdotal data from me, I reached 6000 encounters of the same Pokemon without finding a shiny. And from YouTubers I watch they seem to be having much more luck with the Masuda method. So I'd stick to Masuda for now if I were you.