You have a visibly damaged cap I marked in red, might or might not be an issue. This is a 1.6 board. If it were me I'd just do a standard re-cap. If you have an esr meter...you can test the caps. But these consoles are so old....re-capping them at this point is pretty standard practice even if only as a part of "preventative" maintenance.

Definitely make sure you have the correct laser readings for the correct model. I've seen an increasing number of older disc drives from various devices beginning to fail because the caps, or even resistors, are getting too old and no longer holding their proper value. For example a lot of og xbox disc drives are playing up these days or outright "failing" because their resistor packs have become too resistive, beyond spec and outside of fault tolerance. There are a number of possible causes which are not always the laser, especially now that these devices are so old and well worn. If the laser measures correct, and the motors are all doing their jobs, then perhaps it's time to start checking the cabling, capacitors, and resistors. Even inspect for corrosion or trace rot.

I just had one on my bench recently. It appeared to be overheating, but wasn't. It turned out to be the 0ohm resistors on the lines that relay temperature info. Three out of the six were measuring well over an ohm. They are supposed to read 0.3ohm no higher. In this units case it was old bad solder, fresh solder restored all 6 to 0.3ohms and the false overheat went away. This is a different board. Someone damaged it and tore the pads and traces, ignore that, this is just for illustration purposes. The area circled in red, the yellow lines are where the 0ohm resistors should be. Test those resistors. They should be 0.3ohm, a small margin of error is okay (like say 0.4). If they measure close to or higher than 1 ohm either the solder has gone bad or the resistor has gone bad. They are very small, the testing isn't hard but the work is a pinch difficult/delicate. Anyway overheat issues on 360 s aren't always apu, as I was trying to state before. If you search "xbox 360 S false overheat" you may even find...

You mean the screw heads are stripped? If so there's an old trick that some times works. Take something like a rubber band, place it on the screw head then press the screwdriver into the rubber band. The idea is the rubber band can in some cases compensate for some of the lost/stripped material. If this works, get new screws before reassembling to make life easier in the future. If the worst case scenario happens and you have to break the shell, do it carefully so that you only break the backside part. Those can be more easily replaced.

I believe the hinge mechanism is not cross compatible between iterations. When installing/reinstalling the hinge barrel it should not have required excessive force. It's the angle of the lid in relation to the console that has to be just right. If the angle is correct the hinge barrel will/should pop into place with finger/hand strength. The video looks to me like the something is missing, misaligned, or broken, either way yes clearly incorrect. You will likely need to start over again, double check to ensure that your source for info on how to do it properly is correct. Remember the angle of the top screen in relation to the console body when inserting the hinge barrel is key. There is a spot past "fully opened" where there should be a little bit of play/looseness but what you have there is way too much play.

So either the gfi brick (power cord) has ceased being able to do it's job or (and this is equally likely/possible) it's doing it's job correctly and your actual power supply has developed a dead short. The version 1.0 xbox had a Foxcon power supply which had a design oversight/flaw. The solder joints on the power port could break and cause a dead short which in turn "could" cause a fire. So xbox release free of charge to all customers that GFI enable power cord. It's job is to trip and cut power if it detects a short in the power supply, this is to prevent a possible fire. This is a potentially serious matter for which there isn't likely an "easy" fix. If you want you could try a different "known working" gfi power cord. If it still happens then the fault will definitely be the internal power supply. That would be the easiest place to start. I strongly advise against messing with the internals of either the gfi or the consoles power supply unless you know what you are doing/dealing with.

So...you have what looks like at least 3 vias that are destroyed. That component in pic 3 looks like a diode and yeah it looks toasted. Also in the first pic, right side see the golden number 4 in a circle? If you go a bit up from there it's hard to say for sure but looks like there might be a damaged trace. Diode replacement, verifying and fixing that trace, bypassing/restoring those vias.......all doable, but definitely not a job for a beginner. However!!! She's dead at this point, therefore it's the perfect place for a beginner to start learning! Nothing to lose. I'd start with watching some videos on running enameled wire to repair the via's. If you can manage that part the rest will be easy.

Solder it back on.

Did you plug it into the same outlet that it blew the first cap on? If so maybe test that outlet. If you don't have one already, outlet testers are pretty cheap off the shelf things you can get at homedepot or even walmart. Power supplies :(. I'm in the process of repairing an og xbox Tuscany psu. Some wiseguy replaced the fuse with a wire. The console took a surge and suffered a catastrophic failure. One of the 200v caps exploded and burnt the board and surrounding components. Every time I identify and repair an upstream component it reveals yet another downstream component that has failed or become faulty. If you are willing to go all the way with this you could just break out the multi meter and an ESR meter, start testing components one by one (after its been discharged of course). I mean either....too much is coming in/being let in and the cap pops, or something down stream isn't letting the cap release it's pent up energy and it pops, or you have a dead short on the main power rail and it pops....

Have you tried booting it with the disk drive disconnected? Sometimes...that tick sound is the laser returning home because of a boot failure. When the console is plugged in does the red standby light come on and stay on, or no? There are so many possibilities, trying to nail down the culprit could be "difficult". Did the repair shop do a Syscon read, or even attempt one? Even if it were HDMI or associated components the repair shop should've been able to tell you so, that type of issue is typical/routine stuff.