Syria to UK: 1,500-1,800 miles?
Syria to Saudi and Emirate kingdoms: ~1,000miles
Saudi Arabia: Safe
UAE: Safe
Bahrain: Safe
Qatar: Safe
UAE: Safe
Turkey: Safe enough.
Now apply the same test for Yazidi christians and other minorities. Are they safe in the above countries? I'm gonna answer as you wont: No.
Embassies have the capacity to proves tens of thousands: Yes. Look at India. I can imagine we receive 10-20,000 requests each year from India alone. Googlefu is weak on this.
You are also assuming every one is going to apply to the UK also.
How about to Greece? And how many Syrians are going into the UK illegally? The migration crisis has hardy affected Britain, rather the biggest affected in the EU is Greece, which is next to Turkey, considered unsafe by most countries in the EU.
You're still missing the point that not every refugee travels from the same location and on the same route and that all muslims are not the same.
There is also a massive difference between applications for visas and asylum applications, which by their very nature mean people aren't going to stand around waiting for months/years to get approval, all the while trying to hide from the security forces of their country, or avoid the bombs being dropped on their city.
Yes, asylum seekers and refugees should apply and or stay in the first safe country, and most are. That doesn't mean that those not doing so should immediately be banned and deported. There are a myriad of reasons why they may not have stayed in the first country, only one being economic.
And on the economic reasons, do you really begrudge people leaving camps in Greece/Turkey to try and find somewhere to make some money to send back to their families, so they can eat, and perhaps don't have to marry off their daughters to some old man, just so she can actually get enough food to eat?