Wish cookie consent could be saved at the browser level and websites can just check against my settings instead of asking me every time.

I guess that makes too much sense.

How should a Transparency & Consent String be stored?

In version 1 of the TCF Specifications the consent string was specified to be stored as either a 1st party cookie for service-specific consent or a 3rd party cookie for global consent. In version 2 of the TCF Specifications, the storage mechanism used for service-specific TC Strings is up to a CMP, including any non-cookie storage mechanism. However, global TC Strings must still be stored as cookies under the consensu.org domain.[0]

Pretty much no website uses it.

[0] https://github.com/InteractiveAdvertisingBureau/GDPR-Transpa...