Hungary secured its' borders by erecting a fence and policing its borders. I'm sure that the majority of EU citizens would not want this but as Jagger would sing "you can't always get what you want". At least border checks that people crossing there are indeed EU "citizens" would help, just as a properly policed internal purge of people smugglers within the EU would.
I expect that this Schengen pillar will be one of the last of these to fall but the UK is not a member of the EU so any any costs of abolishing/ suspending it would need to be borne by the EU? What costs are you thinking of anyway and why on earth should the UK need to compensate EU states if Schengen were to be abolished?
I understand the need for compassionate assistance for genuine refugees who have to flee danger (anywhere). To protect them, the UN policy exists and for true refugees, then I agree that safe countries should jointly share the job of accommodating them, at least until their native countries become safe again. But the high percentage of lone men seen arriving in these small boats, trucks and by other means, fairly suggests that these people are not refugees but rather economic migrants.
I am familiar with the main routes and the methods used to move these migrants. The EU should protect its members with stronger external border controls but, because these are weak, that is why Italy, Greece and Spain became so much affected by the initial waves of landings from Africa and the ME. Yes, you can't blame overrun countries for wanting the migrants to move on but you can expect more of the EU to try to manage itself and its external borders better. After the UK, there is little choice for migrants to move on (northward) further and it looks like that it is a popular final destination for many, anyway. EU freedom FoM and Schengen should be EU internal only. The French seem not require a passport or any ID to allow for these non-EU migrants to enter and leave France (as they should) and even use their naval forces to escort these small boats to open sea. The UK has been affected by illegal migration but has not benefitted from it at all, as far as I can see. That is why this should have been one of the main benefits of the UK leaving the EU.
The different path? As I said, it's way too late.