If a stroller or scooter rental company is one of the "preferred" WDW vendors, it will be dropped at bell services at the time you request. (Or earlier) If it isn't then you need to be present to accept it. The same applies for returning them at the end of your trip.
If you do choose to bring it one with you then you can indeed take it through security, and all the way to the gate where they should gate check it at the end of the air bridge and then return it to you in FL at the same location. The trade off with this method being they aren't always delicate when it comes to handling strollers and car seats. I wouldn't risk a good stroller this way, not to mention schlepping car seat (assuming you aren't going to lap hold the whole trip), carryons and stroller all the way to gate is a ton of stuff to carry.
We did 3 trips with our DS between 14 and 26 months, and not once did we bring our stroller, instead choosing to rent one each time. Now admittedly at this point we could have bought a brand BOB for what we have paid in rentals (or maybe it's our next trip that would have hit a break even) and not worried about damage to our primary stroller, but just didn't seem the hassle for you.