In France the supreme court's role isn't tied to the constitution, just the law. They're to dictate which ways the law is to be interpreted and homogenize the decisions by the different lower courts.
We have a different (actually, two...) administrative organs that ensure that the law follows the constitution
FWIW, the U.S. Supreme Court also has this role. People focus more on their constitutional cases because those are the most impactful/famous, but a lot of their workload is just interpreting normal laws, especially ones that have diverged among the various lower courts.
They have proportionately more constitutional cases because a case hitting the SCOTUS needs to be re-appealed at least twice, and the SCOTUS needs to choose to accept the case (4 out of 9 of the judges to be precise). The SCOTUS only takes a few dozen cases a year, so whatever remains tends to be larger issues.
But yes, very few cases are going to be Brown vs. the Board of Education levels of impactful. Perusing some cases of 2023 reveal one such case "simply" being:
>A case in which the Court will decide whether the "serious drug offense" definition in the Armed Career Criminal Act incorporates the federal drug schedules that were in effect at the time of the federal firearm offense or the federal drug schedules that were in effect at the time of the prior state drug offense.
Basically a bookkeeping case due to the different states as of late re-defining "serious drugs"
It deals with lots of constitutional issues, but Congress could basically just defund it and remit constitutional complaints to other venues if they wanted to. This is done in part with FISA and other national security measures. Congress deemed there to be no venue in the courts for a complaint.
- serving as the supreme court for the administrative "order" - as a judge, the CE does not check if laws are constitutionnal
- serving as legal counsel to the government - in which case it does check if bills put forward by the governement (i.e. not laws already adopted) are constitutionnal but this is merely advisory.