[browser][icu] Load full ICU in library tests only when it's necessary

[ilonatommy]

Follow-up for #80421.
It's mostly about reverting change in eng/testing/tests.browser.targets made in 63e5e60 to the state of: 87caebd and making sure it's working fine.

Tagging subscribers to 'arch-wasm': @lewing
See info in area-owners.md if you want to be subscribed.

Issue Details

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Follow-up for #80421.
It's mostly about reverting change in eng/testing/tests.browser.targets made in 63e5e60 to the state of: 87caebd and making sure it's working fine.

Author:	ilonatommy
Assignees:	ilonatommy
Labels:	arch-wasm, area-Infrastructure-libraries
Milestone:	-

Tagging subscribers to this area: @directhex
See info in area-owners.md if you want to be subscribed.

Issue Details

---

Follow-up for #80421.
It's mostly about reverting change in eng/testing/tests.browser.targets made in 63e5e60 to the state of: 87caebd and making sure it's working fine.

Author:	ilonatommy
Assignees:	ilonatommy
Labels:	arch-wasm, area-Infrastructure-mono, in-pr
Milestone:	-