It's worth remembering this summer's Calais crisis really only got going during the ferry blockades by sacked French seamen. They set up blazing road barricades and actually cut the compound fences on a number of occasions, encouraging the incursions. They saw themselves acting directly against the interests of their former employers, Eurotunnel, who were forced to sell their My Ferry Link operations to DFDS by the UK Competition and Markets Authority who objected to Eurotunnel's total market share. DFDS then put a number of the seamen out of work. The seamen, previously of Sea France, were the last vestige of the former SNCF owned part of Sealink who had been operating the vessels on behalf of Eurotunnel as a cooperative.
One of the big problems with a compound of such a vast size (as big as a town) is that once inside it is fairly easy to disappear and hide. Thus, a breach in the fence can be discovered and fixed fairly quickly, but by the time a response gets to the site, if the party entering is small they may have disappeared without a trace. They only reappear later to make a run for the trains or the tunnels. Larger incursions get a bigger and faster response, but whilst the majority might be rounded up and removed there's no guarantee some haven't managed to slip away and hide.
The problem with Calais is not the tunnel or the EU, it is geography. Whether we remain a member or not, a very large proportion of our trade and land travel will always remain concentrated on this one short crossing route by ferry or by the tunnel. There are no other places in Europe where so much legitimate cross-border movement is concentrated into such a narrow corridor, operating 24 hours a day. Most migrants already in Europe wanting specifically to get into the UK have no real option but to congregate around Calais, unlike in Europe where unfenced land borders often stretch for hundreds of km and there is no effective border control between states anyway so there's rarely any kind of build up at any particular site. If someone landing in Italy wants to get to Paris specifically they'll just go there by whatever means and apply for asylum or whatever when they get there. Clearly it's impossible to do that in UK unless at first they get through the security at the ferry ports or tunnel entrance and then risk their life trying to clamber aboard lorries or trains or as in this case actually walking through the tunnel.
According to the infographic animation in the Sky news report an alarm was raised early somewhere around the French crossover cavern (about a third of the way in), but the man was only picked up much later near the UK end. Eurostar traffic performance, as shown by RTT on the day shows no significant delays so presumably only limited cautions or closures were applied, or none at all. Perhaps a widespread search would have required a more disruptive closure when they knew he was bound eventually to pop out at the UK end.