Although speeds were low, 8 trains accelerating together must be hefty load, but probably only lasted under a minute before they were coasting and substation got too warm.
Yes, remembering that from the technical perspective, this type of electrical equipment usually has short-term and continuous ratings.
From a commercial perspective there will be capacity agreements that limit what the railway should draw from each supply point (financial penalties for going over). Generally, a commercial customer will get a capacity agreement in line with their demand, rather than the technical limitations imposed by the capacity of transformers and cables - e.g. if a factory replaces one machine with a new energy efficient machine (or replaces it with a machine in a faraway country with lax labour and environmental laws), then one of the savings is a reduction in cost of capacity for electricity.
This approach also means that if you add a new demand like EV charging it can sometimes be done with a stroke of a pen, or very minor changes to circuit breakers etc. This is increasingly rare however, as the proliferation of renewable generation and EV charging means a lot of this surplus has been used up. As the railway has become more intensively used by longer, heavier, higher-performing trains, then that again is about expensive new grid capacity.
Back to a point the original poster asked about
What if several trains are on the move together? The load on the station's traction power supply must be a big peak.
For the various reasons supplied by others, having a lot of trains diversifies the load and smooths this out and the demand becomes more predictable percentage of the theoretical max. You could see something similar if you look at your own home's demand for electricity, which would appear quite spiky as you turned on things like kettles and toasters versus what it looks like if you aggregate many houses together where you just kind of get very smooth rising and falling demand. At the other end of the scale, if you imagine a very rural bit of electrification, you might size it on the basis of only 1 train being in that section at a time. The train leaves and demand goes almost instantly from a low level to around 100% of the theoretical demand you've sized for, which might be more of an issue, especially if you then decide to replace something like a Cl 319 (~1 MW) with a 350 (~2 MW), or double the train length etc.