The Undermine

"Posting in the comments isn’t the ideal place to maximize the number of people who will see this." is very true, but you got lucky this time. :)
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"What is the general approach?"

There are two possible approaches. The easier of the two is to go to a Family page like this, or another like Legion https://www.wow-petguide.com/Section/17/Family_Familiar and click the Blue "All Families" button. Then just load and Import the whole Rematch scripts you get. And go and do them. Simples.

But ......

While you might have "enough level 25 pets of sufficient variety and number as far as I can tell", you may fall short on some Breeds. You have obviously not been paying a great deal of attention to getting the most useful Breeds for battling on the most used species. This is not a huge problem, but you may find yourself going out to hunt for specific Breeds for some specific battles.

I checked a couple of common pets in your collection, and I see you don't have S/S Breeds for Nether Faerie Dragon and Qiraji Guardling. Both are very useful pets, and S/S is the most required Breed for both. I have a thread about that here: https://eu.forums.blizzard.com/en/wow/t/pet-breed-survey-2019/20349

Xu-Fu's amazing strategy picker will try to work around that, but if you find yourself with an unsatisfactory strategy, check if there is a higher-rated one using a Breed (or statistic requirement) that you don't have.
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"Will Rematch slow down or die if I load it with too many teams?"

Rematch will be fine. I have A LOT of teams, far more than you will ever load, and Rematch handles them all easily.
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"Do people run the scripts associated with each of the family battles "

Some do. I don't and won't use scripts at all, but I certainly would not use them for one-off battles.
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"insights to share about the process?"

Start with BfA; It's the easiest.
Get the addon Family Familiar Helper.
Start Legion ASAP. All of the other battles are available to try 24/7/365. Legion battles are available only when their WQ is up, so you WILL be waiting and checking WQ every day for a month or so.
Remember that for most of the Family Achievements, YOU DON'T NEED TO COMPLETE THEM ALL. There are a few encounter/Family battles that can be frustrating. Feel free to skip where you can.
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"forum or discord where people discuss these sorts of issues?"

Both US and EU Blizzard official forums have a Pet Battles forum. Xu-Fu and WarcraftPets.com both have Discords. Xu-Fu likk is on the home page. WCP link is under Community Menu->Discord Server. I'll see you there! :)
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