The only way other way to get priority is by bettering your price. If minimum tick size is set to a very low threshold (or no threshold at all), "outbidding" someone for a better place by offering a economically insignificant price improvement (say $0.000000001 per share) would become the easiest way to get price-time priority on someone, and hence the value of placing a standing limit order would be reduced: Anyone can just outbid you for a fraction of a fraction of a penny at any time, so it becomes much more likely that if your limit order gets filled, the price will then move against you, and otherwise, your limit order will never get filled. This problem exists today but would probably be made worse by a smaller tick size - in fact I believe it's one reason why liquidity rebates are offered.
1. https://www.bloomberg.com/opinion/articles/2015-01-13/hide-n...