Mr. OldPilot.
You are incorrect! Let me give you some background. I am an Air Traffic Controller at KFLL(That is Ft Lauderdale International Airport). I have been a controller for 10 years. I have a unique understanding of the National Airspace System.
At its most basic. Every tower owns all the airspace depicted on the charts. In the case of FLL, that would be surface to 4000 for 5 miles then 1200 to 4000 upto 10 miles. Through a Letter of Agreement(LOA) the airspace can be delegated to another facility(i.e. another tower or appch) for control authority. At FLL we have delegated all airspace above 1200 to MIA APPCH. This means that any transition below 1200 and within 5 miles would need to call the tower directly.
Why is this so hard to grasp. Authority for sequencing is also the towers responsible unless delegated to an appch by LOA.
Here is the difinition from the FAA7110.65
3. CLASS C− Generally, that airspace from the surface to 4,000 feet above the airport elevation (charted in MSL) surrounding those airports that have an operational control tower, are serviced by a radar approach control, and that have a certain number of IFR operations or passenger enplanements. Although the configuration of each Class C area is individually tailored, the airspace usually consists of a surface area with a 5 nautical mile (NM) radius, a circle with a 10NM radius that extends no lower than 1,200 feet up to 4,000 feet above the airport elevation and an outer area that is not charted. Each person must establish two-way radio communications with the ATC facility providing air traffic services prior to entering the airspace and thereafter maintain those communications while within the airspace. VFR aircraft are only separated from IFR aircraft within the airspace.
Notice the phrase "Although the configuration of each Class C area is individually tailored".
Usually sequencing to a Class C airport is handled by an appch control. Low altitude transitions are not always controlled by the appch. Could you imagine an appch control working a helo 500agl .25 miles off the departure end of a runway and the tower controlling the departures. This does not happen. The tower should be working both a/c.
That is the end of my beating of the dead horse!
You are not correct for every airport.
Thanks.
P.s. All of the Class D airports around KFLL are also not as you say! HWO, FXE, OPF, and PMP all have given up their airspace above 1500 to MIA APPCH. So you will have to call appch for those Class D airports for entry at or abv 2000