Hello,
I watched the first video. You'll have to elaborate on the bits you don't understand or I'll just be guessing. It all looks fine TBH, though not everything was shown in the video, and these are likely the bits that explain more.
I'll take a punt at 2 things being problematic if the whole story is not being told, and what I think is going on.
The crown moulding on the top case is going across grain to the carcase sides. If it was fixed it would be a problem but it is not shown how it is attached in the video. I'll bet a pound to a pinch, it is not glued, but slid onto dovetailed keys, so the moulding will effectively slide as the carcase sides expand and contract seasonally.
The bottom case doesn't have this problem as the case side is orientated with the long grain running front to back. The moulding can be glued here. The seasonal movement will be vertically in the lower case. The only problem here is the fit of the drawers, but that is accounted for by leaving the drawer sides with a fair bit of room for the carcase to move vertically around them. The drawer fronts are rebated all round, essentially an overlay, to hide the larger than (aesthetically) optimal gaps at the top. There could be a problem where the lower case sides meet the stiles, but again there is no detail as to how they are fixed. Could be sliding dovetails. However, the time when this piece was originally made, it might just have been nailed. There is a bit of movement tolerance with nailing, especially with walnut which has a very small movement in service. You will probably have noticed how many antiques have splits in the sides, though! It may well be that the makers just followed the old ways and it will just crack. Believe it or not, I have heard it said that that is the expectation and it is fine, it is a verbatim copy because the client wanted it to be so.
I didn't look at the second video, but saw the first 20 seconds. As far as I saw, the carcase looks fine.
What are you having trouble with?
Mike.