ZMA Controller reported being inundated with traffic during a session with weather.

Narrative

I took over as the R-side at sector XX at XA:18 Zulu. Huge push of traffic coming out of Miami approach and all deviating. The EDST at sector XX was full with a scroll bar from how many airplanes there were. This is not good especially with sector XY open taking off the chunk of airspace FL360 and above. So from FL240-350 to have a scroll bar in EDST is not good. I vocalized in the area that this was not safe and we needed it slowed down; I was too busy with the traffic to hear what their solutions were or if there even were any. The ERIDS in the area showed the sector RED and I know that may not always be accurate but a 50 in red will let you know that the sector is probably in trouble.Seems like TMU or Command C enter is afraid to ground stop aircraft to prevent delays; but loss of life isn't a concern. This session we had deviating aircraft with no ability to even see data blocks. My opinion is if we ground stop or put massive in trail restriction to ensure the safety of aircraft; then passengers won't mind 45 min delays.