Some grant proposals mention that they only want funds for students and no funds for faculty such as summer salaries. Would this point increase the chances of being funded by some agencies? Specifically, if two similar competitive projects were proposed and one asked for faculty salaries and the other didn't, would this make the difference?
This of course depends on the kind of competition, agency, country, etc. Those agencies and competitions I'm familiar with, always rank applications based on merit and not on financial need.
There is no clear answer. It depends.
If funds are available for any of the two, the former is likely to win. Giving money to the faculty raises the chances that the faculty would be actively contributing.
If the money is available for only one project, then there is no simple answer: the former may have more impact, while the latter may be cheaper.