Hi and thanks for any thoughts on this!

I have a 2002 Silverado 1500 4.8L v8 (110k miles) that sometimes has trouble getting moving from a stop. It happens sporadically (both hot and cold) and will sometime occur from every stop one day, but then not happen again for a few days.

From a stop, I'll apply gas and it'll get to around 1500rpm but not go more than 5mph at first. Most of the time if I keep applying more gas, it'll eventually climb to around 10mph and later shift hard to get to 2nd and start going normally. Other times I have to shift down to 1st manually to get going and then shift back to Drive after I'm moving. It sometimes makes a bit of a rattling noise when it's happening, but most of the time it's just the revving noise that makes it sound like it's picking up speed but it's not. I also sometimes notice it shifting hard when coming to a stop, so I wasn't sure if it's maybe just not downshifting properly and then starting in a higher gear than it should—just a guess though, as I'm not a mechanic.

I've cleaned the throttle body and mass airflow sensor, but it didn't help much if at all. I've had it to two transmission shops, but it wouldn't do the slow take off while they had it driving it around, so they said it's running great. They also did diagnostics both times without any issues showing. I plan to get a transmission fluid and filter change, but I was hoping I could get some other ideas of what they might check before taking it back to the shop. They said they can tear the transmission apart or replace it, but I want to make sure there isn't something outside the transmission that might be screwing things up before we start rebuilding or replacing it.

I'm attaching a video to show the speed and RPM. In the video, I'm trying to leave from a stop and giving it a lot of gas—enough that it'd have usually gone to 20mph or more quickly. Instead it goes from 1000-1500rpm and between 5-10mph before it finally gets going faster and eventually shifts.

Thanks in advance for any thoughts on this!