That isn't why your rent is up $500 this year, but LA definitely has a supply crisis!
Cedillo was in charge of the city council's housing committee, which was implicated in a lot of pay-for-play dynamics and two past members are now in federal prison. I'd argue that intentionally creating a bottleneck to increase payouts form developers did a lot more to cause the housing crisis than somebody who isn't even in office yet but YMMV.
Hernandez will be fine.
In general, progressives don't halt construction. Most of the time, neighborhood councils, which trend conservative or centrist, are the ones invoking CEQA and showing up to planning commission meetings where these decisions are made. These aren't progressives making these public comments, sending these emails, and blocking new development. These are liberal, moderate, and conservative homeowners, by and large, and usually older folks at that (since young people can't afford to buy million+ dollar homes in LA).
The only development that progressives typically block are that don't adhere to mandatory inclusionary zoning minimums for affordable units. (LA doesn't have the same aggressive left-leaning NIMBYism as the Bay Area, imo). The 725 unit complex across the street should have 73 affordable units, which would actually do a lot to alleviate the supply crisis in lower segments of the rental market. If you come to my side of town, I can show you a lot of new construction that has added upwards of 1000 units of affordable housing to the rental market over the last few years.
The narrative that you're embracing sounds a lot like what I hear from developers, but, speaking as a subject-level expert in this area, I really don't think that it's true. Happy to discuss more over PM, but if you're actually interested in addressing the housing crisis and not just in embracing reactionary politics, then you should take a look at the United to House LA ballot measure and the Our Future LA coalition, which represent, in my opinion, two incredibly hopeful moments in an otherwise pretty depressing field of supply-side housing policy.