I have some ZX Spectrums, and in some of them I observe a kind of "disintegration" process of the main board. The connectors break away etc. During the tests one of the boards burned out :(.
How can I stop or slow down this ageing process?
The ULA chip in the ZX Spectrum (48K version at least) has been completely (100%) reversed engineered by Chris Smith. He released the design to the public (look up his book...it's amazing) and since then, people have created new boards that will fit in a ZX Spectrum case. These boards (along with off-the-shelf components) can recreate the functionality of the ZX Spectrum.
Also, the keyboard membrane, keyboard connector and case have all been remade recently. You can literally build a new ZX Spectrum in all kinds of colors and run 100% of the software (or so I've read). All without emulation! This is pretty amazing in my opinion.
Those boards are on eBay. Look for "Harlequin".
One example is here:
That site uses a 3D printed case but since then, there have been professional cases made using the original molds.
Now, if creating a modern version of the ZX Spectrum is not your cup of tea, then there is no reason why you can't recap the board and "future proof" it as best you can. The ULA is about the only chip you need to worry about. The others can be replaced pretty easy and there are tons of new equipment made for it.
I suggest watching these videos on repairing ZX Spectrum's.
He's pretty much the expert on repairing ZX Spectrum's. Also, he sales (or at least he used to) a modern SRAM replacement for 16K/48K Speccy's.
This answer is not strictly related to your question: you can re-create PCB (nowadays PCB manufacturing is cheap) and then just resolder key components on the newer one (also replacing such things as electrolytic capacitors). If you already have a broken ZX, accurate soldering off of all components could be made, then the empty PCB could be both sides scanned -- like it is a piece of paper. After that, newer PCB drawing in some CAD would be next to trivial.
Preserving PCBs of that age is possible. Firstly, replace any components that might leak, such as capacitors. Plastic parts such as IC sockets and connectors can be replaced too. Then you can move on to the actual PCB itself.
The PCB is made up of multiple layers. The most important one is the copper layer which carries electrical signals. It is protected by a top coating that stops it oxidizing and degrading, but the coating itself can get damaged or degrade over time.
It is possible to replace both the copper and upper layers with some work. The copper layer can be fixed by cutting strips of thin copper to patch damaged tracks. The solder resist layer on top of it can also be patched up manually.
Actually, there isn't much on a ZX Spectrum mainboard that would require specific care or couldn't be replaced with parts still available. The ZX Spectrum is simple enough to be easily replaced and all major parts are still available, so, if we're talking decades instead of centuries, the situation is quite good (The ZX Spectrum was made in such large quantities that, beyond off-the-shelf replacement mentioned in this answer, you can easily find a donor for parts. I'm not mentioning this because this would be my last resort for repair - Every ZX Spectrum which is still around deserves to be preserved). The only things that actually "age" significantly are the capacitors on the board and anything made from plastic:
So, from an electronics point of view, the replacement situation is quite good, very probably better than with any other 8-bit computer. Mechanical components (especially the keyboard connectors) are a bit harder to replace, even if complete case and keyboard replacements are manufactured in China. Plastics are aging, and replacements have become rarer. An easy way to preserve a ZX Spectrum for a long time is to store it in proper conditions. Plastics are sensitive to extreme heat and UV (sunlight), the capacitors age much faster under temperature, the metal parts are allergic to humidity and might begin to corrode. Store the computer in a dry, cool and dark place and use a replica, like the Harlequin, as an everyday machine.