So, I've got a decent number of BTF-Lighting strips and they've behaved. Obviously there's always room for duds. You can also get the same LEDs for slightly cheaper but more slowly if you go to their AliExpress store.

The strips you've got are not intelligent strips, so everything will be the same color. But the intelligent strips tend to have a worse CRI, plus the wiring's actually a bit simpler this way, so this is probably a fine direction to go. If they were intelligent strips, you'd have to connect the output end of one strip to the input end of the next. With these strips, you can just wire them up in parallel with one set of wires for each strip using a Y connector or whatnot.

I put some LED strips on my shelf lately. What I ended up doing was soldering the wires directly to the strips after cutting them, and then I designed and 3D printed an integrated clip-diffuser to hold them in place.

The nice thing about the COB strips is that they are already quite diffuse without doing much of anything, so honestly you might just be able to peel the tape off the adhesive on the back and slap 'em on the shelves.

I guess you could try these, if you aren't going to be wanting to solder the wires: https://www.amazon.com/BTF-LIGHTING-Transparent-Connector-Solderless-Extension/dp/B0BNKXZFYB?ref_=ast_sto_dp&th=1 but I've never tried it. The spacing for the COB strips is a bit odd compared to the older strips so a lot of the options won't work.

Pretty much any wires will work, but if you are going BTF lighting, you can just get some of this: https://www.amazon.com/BTF-LIGHTING-Electrical-Extension-Connection-Changing/dp/B07SHGYC27?ref_=ast_sto_dp&th=1 You can just use whatever random stranded wire is hanging around that's between 18 and 22 gauge.

The sort of controller you'd probably want, which I've never used because I just make my own controllers, is something from this category: https://www.amazon.com/stores/page/1C41EF09-3C83-4905-A210-06C1760251A5?ingress=2&visitId=a4e52e7d-5670-460d-b0e6-2d14a400f93c&ref_=ast_bln

But then you'd want a power supply that matches the strips (either 12V or 24V) to go with the controller. The 24V strip is probably more optimal if you are doing a much larger LED project but 12V will do fine.

Looking at the end, there's screw terminals so you might just be able to fit all three sets of wires into the screw terminals. Or you can use a terminal block. I don't really have a Y connector type that I really like, but pretty much anything will do there. Wago lever lock connectors might work or wire nuts for example.