Not necessarily - the heatsink could be clogged, dislodged, the fan could be toast, or the thermal compound could be dried out and dead.

Yes check all your thermal paste and thermal pads for sure.

OK, well, the laptop cpu can run 60C hotter than ambient, or more. So, if the AC cools the room down to 68, and the cooling pad gives you another 5C drop, it really doesn't cool the inside of the laptop by much. You really have to blast the computer with cold air. I also remember HP GPUs getting so hot that the solder balls on the BGA melted and the GPU lost some of it's connections to the array when the machine warmed up. That is a difficult repair because it requires heating the GPU to 200C, and monitoring it.

Before I started using the cooling pad, my lower-end laptops would overheat and crash after an hour or two, but after I began using the pad, no more problem. The Linux Mint "Disks" app always shows "Disk is okay" regarding the temperature. The cooling pad does make a difference. The only exception is my Dell Latitude D820, which I'm running right now. Although it's over 10 years old, it's still in great shape and has an excellent internal cooling system so it doesn't need the cooling pad.

Update on Dell Latitude E5410 laptop problem with black screen after bootup: Apparently it wasn't a graphics card problem after all. I came to remember that the problem started soon after I had upgraded the RAM from 4 to 8 GB's. About a week ago, out of curiosity I removed the 8GB ram stick and reinstalled the 4GB that came with the laptop. Bingo! The computer boots up again. All's back to normal. According to the product description on Amazon, the 8GB RAM stick is supposed to be compatible with my laptop, so it's probably a defective module. I'm going to test it with another laptop and see if the same thing happens.