I will say that Elite Guard probably gives the best bang for your buck on eGatcha tickets. For gambit it seems like you need to get all of the slayer cards to even have a chance while with Elite Guard you can call on friends to help you with SEGs. It's also easier to clear out all of the point rewards in Elite Guard as opposed to Pero Gambit (2 million trophies? seriously?). Pero Colo is also quite simple as long as you play it correctly; however, it really is a grindfest and I mostly only play for the colosseum gatcha anyway.

Pros(+) and cons(-) of each:
Pero Gambit:
+ It is the least taxing of your time (you can wait for your stamina bar to refresh before continuing)
- It is very hard for a free player to get a good rank
- You don't get as many items because a lot of gold and silver cases expire

Elite Guard:
+ Every battle is worth it in terms of items dropped and event points
+ The slayer cards really help you no matter how far you are in the game
+ You only need to tap an SEG to get credit for the kill
- It is the most taxing of your time as you need to wait for your focus and time to spin up a guard
- You're not killing a SEG without a friend, online, that can take it out in one shot

Pero Colo:
+ Every battle is worth it in terms of event points and colosseum gatcha points
+ Even low level players can rack up a good score
- Very taxing of your time
- Requires older, unavailable cards (such as Lillith Lil) to to well
- Unless you can get each SR and R slayer card and fully level them, playing with them is really not a good idea
+/- Princess can be a good way for low level and free players to get an SR, but anyone that doesn't vote for the top princess is permanently locked out of that card
+/- So many pero shakes that are only useful for leveling up