Sounds like you broke a socket or wire when replacing the lamps (bulbs). This could have happened with any lamp, it was not the fault of the LEDs.
Try to reset the breaker with no lamps in the fixture at all. Then put in one lamp at a time and try again each time.
Either way, I'd replace the fixture. If that is not possible financially I would remove the fixture and find the bad wire/socket and remove it.
LED strips could certainly light up a room, but they're unconventional.
I think to get good light coverage, you'd have a few strips in the middle of the ceiling. Whether this would look odd is for you to decide. (Personally, I'd not do it.)
Practically speaking, you'll probably need to have transformers somewhere, so that's a concern.
If you want dimming capability, you'll have to choose your strips for that application.
There are a lot of poor quality strips/transformers out there, so unless you want to make a hobby out of replacing your lights, be careful purchasing.
This varies from product to product, but if you use a long string, you might experience voltage drop from end to end. To prevent this, feed the (+) to one end and the (-) to the other end with a reasonably substantial conductor.
Again, widely variable, but heat buildup is the enemy of LEDs. If you can stick the strips to something metal, that'll help.
On the plus side, strips are relatively cheap, so you could do it and see what you think.
Best Answer
You're not allowed to use fixtures where some fool could remove the LEDs and install incandescents. Hence if you use swappable "bulbs", the lamp socket must be a unique type made exclusively for efficient lighting. This is done for two reasons: