Why does Urza's Saga keep its activated abilities under Blood Moon?
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The short answer
Urza's Saga's chapter-granted abilities survive Blood Moon because the effects that grant those abilities operate in Layer 6 (ability-adding), which is always applied after Layer 4 (type-changing, where Blood Moon operates). This is determined by layer sequencing, not by timestamp priority.
The cards
Urza's Saga
(As this Saga enters and after your draw step, add a lore counter. Sacrifice after III.)
I — This Saga gains ": Add ."
II — This Saga gains ", : Create a 0/0 colorless Construct artifact creature token with 'This token gets +1/+1 for each artifact you control.'"
III — Search your library for an artifact card with mana cost or , put it onto the battlefield, then shuffle.
Blood Moon
Nonbasic lands are Mountains.