The real problem was not just the movement of Egyptian forces in the Sinai, it was the closure of the Straits of Tiran. By doing this, which was legally a causus belli, in combination with existing limits on the Suez Canal (no Israeli ships or ships bound for Israel), a huge burden was placed on Israeli access to Maritime trade. It meant anything from south or east of Israel would need to circumnavigate Africa, and then transit the Med to get to Israel rather than going to Eilat. If the straits are not closed the war might be avoided. However the issue of more forces in the Sinai cannot be ignored, Israel's problem, especially in 1967, is a complete lack of strategic depth so the practicality of a tactical retreat and a counterattack is limited. Furthermore because the Israeli Army is highly dependent on mobilized reserves, with the effect on the civilian economy, mobilizing in response to the threat and staying mobilized fro a prolonged period is also not very practical.
If there is no war in 1967, but one starts later on, depending on the timing Israel may by then have a number deployable nuclear weapons (not just one or two) and if the Arabs are doing better and given their start lines will be the 1967 borders, not the 1973 ones, those nukes might very well get used tactically and/or strategically.