Is there a specific reason for having a very short window for passengers to get to the boarding gate before closing it?

Normal airlines will have a period of "Boarding" before "Last call": Ryanair's first call is also a "Last call". The gate probably actually closed at something closer to 20:40. In other words, the window isn't as short as they pretend, but they want to put pressure on the customers.

I mostly see this for low-cost airlines. In that case, most passengers will have been processed before the aircraft even touches the ground. The gate will already be closed when passengers from the previous flight finish to disembark. So yes, they can and do change the gate at the last minute.

But in some cases, it seem to go beyond that. Announcing late puts pressure on the passengers to all show up at the same time and occurs systematically in some places (e.g. low-cost terminal at Schiphol airport near Amsterdam). And at AMS, the gate really does not open before it's announced so it only stays open for 15-20 min tops. It's all part of an effort to minimise turn-around time and reduce costs, even at the expense of comfort.

By contrast, bar some unexpected issue, at Schiphol's other terminals, gates for long-haul flights from legacy airlines will be announced well in advance and stay open for a longer period of time. In that case, the aircraft will typically already be at the gate when you arrive, which is not the case for low-cost operations.

In this particular case, it seems that Glasgow airport has a dedicated “low-cost” pier and I note that only Ryanair and Easyjet have flights departing from gates that are 7-8 min away from the place where you took the pictures.

I suspect some airports charge these low-cost carriers lower fees to keep their traffic but deliberately inconvenience passengers so that the low-cost pier only appeals to people willing to trade comfort for price, while other airlines keep using the more expensive regular pier and have gates that do not require so much walking.

If your problem is that your grandmother cannot walk quickly, consider asking for Boarding Assistance for her and her companion.

Even the budget airlines have a plan for providing boarding assistance. Sometimes it includes a special bus to take you to the plane (not the gate) and a special lift to put you into the plane by the back door, before any of the other customers are allowed on.

In theory only 1 companion is permitted, but my family had success with 1 companion and 1 child accompanying the person who had recently had knee surgery…

In my experience, boarding gates will stay open (for a reasonable amount of time) until all passenger who got boarding passes come for boarding. Several times, I would arrive as late as 15-20 minutes past boarding time.

I don't advise you to be late for boarding on purpose, but I don't think your grandmother should not travel with RyanAir because she can't run fast enough. They will wait for her. Denying boarding to someone who has trouble walking (or just went to the toilet for 5 minutes) would result in endless influx of customer complaints, ruining the business.

In aviation everything is optimised to the maximum, as long as it doesn't undermine safety, and Ryanair is especially notorious for this.

Most of the time the planes come down, unload cargo/pax, refuel, sometimes crews change, then in shortest possible time they load in passengers and cargo, then fly away. All that is done on a really tight schedule. That's why passengers are advised to be at the gate before the boarding starts.

As for late announcement - well, we're all human, including the people announcing things in aieports.

Source: work experience at an airport

In addition to all of the great answers above, one thing worth saying about low cost carriers in general (and Ryanair specifically) is that minimising turnaround time is absolutely crucial to their business. While we don't know exactly how much Ryanair's fleet of 737 aircraft cost to run per hour, it's a lot (an upper bound would probably be on the order of $11k USD per hour as you can hire one for that much), we do know that it's going to be earning them exactly zero money sat on the ground.

All airlines are subject to, more or less, the same fuel costs, use essentially the same aircraft (which are always chosen based on their economic efficiency for a given route, which is not necessarily the same thing as their temporal efficiency -- in Japan you find all-economy 747s doing short haul, for example) and have essentially the same (legally mandated) service costs. Turnaround time is basically the only thing that is left for them to compete on.

There has been much academic analysis done about minimising turnaround time; e.g. this article details how operations research has been used to look at the problem. Aviation throws up many interesting problems like this -- several of which are NP hard, and Boeing themselves have produced a document detailing how a time saving on the order of minutes can result in the difference between making a lot of money on an airline route and being faced with some very large shortfalls to make up. Heck, a quick google shows up this Masters' thesis looking at Rynair's profitability and turnaround time. If you'll forgive me selectively quoting the relevant bits about just how crucial turnaround time is to Rynair, they're quite telling:

By taking about half the [turnaround] time of the larger airlines like British Airways (BA) or Lufthansa, Ryanair’s planes made an average of nine trips per day as against the average six of larger airlines.

[...]

No other low-cost airline managed to replicate Ryanair’s results. According to analysts, its “cost per available seat mile” (the yardstick used by the airline industry to measure costs) was 30% lower than the average for Europe’s major airlines, and its productivity - as measured by the number of passengers per employee - was 40% higher. As a result, Ryanair could break even when its planes were just over half-full.

I would say that the fact that Ryanair does not tell you which gate to go to ahead of time is because they themselves do not necessarily know. When you add in the fact that Ryanair pays tiny landing fees to most airports (negotiated on 15-20 year terms), it's entirely likely that their gate -- which they might want to squeeze in around other, longer operators -- may be necessarily uncertain until it is clear what has happened elsewhere in the airport.

What's far more important to them is, frankly, that the aircraft gets to a gate and leaves with some (not necessarily all!) passengers as soon as possible. I'm afraid that, as far as they're concerned, you and your grandmother's inability to get to the gate on time would just be another willing sacrifice on the altar of capitalism.

I saw in a documentary that the reason is commercial. Many European airports want you to be on the center, near shopping area, so you can spend more. They get much money from passengers. In US airports are public, so they exploit less passengers.

For such reason, they try to keep security queue as short as possible, and try to get as quick as possible passengers on the central part of airport.

But delaying embarkation has also costs, so you get the time when the will announce the gate, and usually around 45 minutes before departure.

If you look around or you check on some other online sources, you may see the gate before it is published on airport screens. Sometime it is also on the ticket.

Just I found confusing that now you may find boarding time and not departure time on your ticket (or on screens), so it is more difficult to find your flight on the list.