Electronic – Is it safe to use this relay with mains voltage

Metal strips are used in swtichboards in my country because they can handle more power and have less risk of catching fire (cheaper maintenance, less human error risk). They are also cheaper to manufacture, since the strips are connected and held in place by screws and not soldering. Soldering a wire to a metal strip is not especially easy, though, and there's a risk of not doing a great job with one wire and it coming off and touching something else.

The ideal way, in my opinion, would be to use some sort of thimbles which you can hold together with nut and bolt. If you're planning on making a PCB, you could use the PCB itself to distribute power with ultra wide traces (not a good idea if you want to control heavy duty electronics like ACs or room heaters). The PCB would give you the freedom to leave large pins for the thicker mains wire to be soldered properly.

The optimal solution without a PCB or buying new components like thimbles, in my opinion, would be your option two where you use the caps as thimbles. I would advise caution, though - the outer surface of the the caps are also conducting and therefore these would have to be well secured and insulated. Remember to not skimp on the wire thickness to accomodate more wires per cap - that could lead to burning plastic, broken wires, and worst case, electrical fires. A fuse may be a good idea to include in your box, to be safe.

You say the push buttons are non-existent? I'm assuming they are rubber/graphite buttons, which work just like push buttons. I'm also assuming that the buttons pass very small currents and voltages. You want to measure to make sure.

In either case, yes, you can simply replace them with mechanical relays. Of course you want to use relays which work at 24V coil voltage. Size wise, reed relays are pretty small, but you can also find SMD relays that are small that you could solder without issues as well. The diode is to protect your output if DC. If AC, you should get a relay meant for AC coil voltage instead of DC.

^{simulate this circuit – Schematic created using CircuitLab}