How exactly the phase of contact worked out and how the intervals and how they were reinforced or the intervals closed is not really clear. Livy recounts how it might have happened:

when an army had been marshalled in this fashion, the hastati were the first of all to engage. if the hastati were unable to defeat the enemy, they retreated slowly and were received into the intervals between the companies of the principes. The principes then took up the fighting and the hastati followed them. [10] The triarii knelt beneath their banners, with the left leg advanced, having their shields leaning against their shoulders and their spears thrust into the ground and pointing obliquely upwards, as if their battle —line were fortified with a bristling palisade. [11] if the principes, too, were unsuccessful in their fight, they fell back slowly from the battle —line on the friarii. (From this arose the adage, “to have come to the triarii,” when things are going badly.) [12] The triarii, rising up after they had received the principes and hastati into the intervals between their companies, would at once draw their companies together and close the lanes, as it were; then, with no more reserves behind to count on, they would charge the enemy in one compact array. [13] this was a thing exeedingly disheartening to the enemy, who, pursuing those whom they supposed they had conquered, all at once beheld a new line rising up, with augmented numbers. [Liv. VIII, 8, 9ff.]

Livius however writes long after that time - the literary evidence for these early wars and battles is very shaky.

Be that as it may, you're right, each interval in the manipular order offers an opportunity for the enemy to engage the flank of the maniple (though he also opens himself to attacks from three sides himself). This is why the maniples probably were deployed with much smaller intervals, who closed themselves upon melee contact - either by fanning out to the sides until contact was achieved, or having one of the two centuries per maniple in reserve to move out to close the gap. Should intervals open again during the fighting, they could be closed by one of the maniples of the Principes, and, should it come to that, the Triarii.

However, the advantages of the manipular checkerboard formation are evident. It allowed a more fluid style of combat, where units or even individuals could surge forward to release their missiles, and easily fall back to the main line. The Velites could also easily move through the intervals once they had released all their missiles. It also made the process of reinforcing very easy, since the relevant units were already where they would be needed. This ease of reinforcement was a considerable advantage, since you could introduce fresh units into the line when your enemy, deployed as a single line, becomes exhausted - something the Etruscans learned to their sorrow at Sutrium:

The Romans were equally prompt to begin the battle. The two armies rushed together with great fury, the enemy having a superiority in numbers, the Romans in bravery. Victory hung in the balance and many perished on both sides, including all the bravest, and the event was not decided until the Roman second line came up with undiminished vigour to relieve their exhausted comrades in the first; and the Etruscans, whose fighting line was supported by no fresh reserves, all fell in front of their standards and around them. [Livius IX, 32 8f.]

All in all, the manipular "checkerboard" is a logical evolution of the Phalanx. It is incredibly hard to march a line of men across terrain and keep them straight, so there will be small intervals opening up as some are faster, some lag a bit behind, have to step around or climb over obstacles and so on. These intervals have to be kept small, though, and can't be allowed to open up into bigger ones - the advantage of the Phalanx is the security you provide to your neighbor and are provided by yours in turn. If that fails, the Phalanx can fail, too, and the fear of that was taken into account. Xenophon remarks in the Anabasis, when faced with deploying the Phalanx in difficult terrain:

“For the line,” he continued, “will be broken up at once; for we shall find the mountain hard to traverse at some points and easy at others; and the immediate result will be discouragement, when men who are formed in line of battle see the line broken up. [Anab. IV.8, 10]

This is one problem that the manipular checkerboard solves. But it requires a high degree of discipline, and trust. Yes, if you stay with your maniple, an interval opening to your side is no problem - as long as you can trust that your comrades behind you will close it at the right moment. This also requires a high degree of leadership experience, both in the senior and the junior officers - all in all, it requires a high degree of military expertise. In the words of the great Delbrück, the Phalanx had been given joints. The separation both along the longitudinal and the latitudinal axis allows the battle line to react to all possible changes in the flow of combat.