Cards like The league of uniform nomenclature, White elephant's gift, and likely Tyrant's Throes cannot be used by tributing/using the Burgesstoma cards (and likely cards like The Original Monarch and the Phantom Knights, but I haven't tested)

I am not sure if it's necessary as a ruling that these "monsters" must retain their monster status post cost-paying, or if it's just that they're not properly coded as normal monsters, given that the costs are incurred on the field, unlike floaters which activate in the grave

This is on the chrome web version

Well in the case of League of Uniform Nomenclature. Burgesstoma cards (and any other trap monsters), only become monsters once they are activated.

Whilst in the deck, they are trap cards, so Uniform Nomeclature won't work.

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As for White Elephant's Gift, it requires you to "Send 1 face-up non-Effect Monster you control to the Graveyard" as its cost (meaning that the cost is only considered paid if the monster being used for the cost does indeed go to the Graveyard). Because of the "banish it when it leaves the field" condition on the Burgesstomas, you can't properly pay the cost with them (barring something like Imperial Iron Wall being active). You should have no problem using these cards for the cost of Tyrant's Throes, since that cost is simply a Tribute (meaning that it doesn't care where the Tributed monster actually ends up).

I'd concede about White Elephant's gift, but the way The League of Uniform Nomenclature is worded makes me uncertain about its use, given that Magical Hats is capable of summoning non-monster cards as well.

This is purely semantics but

After activation of the burgestoma cards, they are treated as normal monsters.

The League of Uniform Nomenclature targets a normal monster, then "summons... as many cards as possible with the same name as the selected card"

because it says "as many cards as possible", and not "as many monsters as possible"

Given today's pendulum world carefully distinct pendulum monsters in pendulum zones as continuous magic cards, in monster zone as "monsters" , and archetype searchers/support carefully word as targeting the archetype "cards" if they're capable of affecting pendulum monsters in pendulum zones, it seems to me that the wording allows summoning based on name, irrelevant of the card type. The same goes for Inferno Reckless Summon.

I do understand that in the deck it exists as a trap card which might void its "nature to be summonnable" ala Reasoning, but still its something worth debating

This always bugs me the same way that tokens cannot be treated as xyz material in that a game should be as self-explanatory as possible and yugioh has developed into this cesspool of contradiction the more they introduce new powercreep mechanics as they are wont to do

"Even if "Stronghold the Moving Fortress" or "Metal Reflect Slime" that is being treated as a Machine-Type by "DNA Surgery" are treated as Monster Cards by their own effects, they are still treated as Trap Cards in the Deck. Therefore, since you cannot Special Summon any monster with the same name from your Deck, "Machine Duplication" cannot be activated by targeting "Stronghold the Moving Fortress" or "Metal Reflect Slime"."

While not the same cards specifically, the same ruling applies.

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