Would the Elo rating system remain useful if different tournament formats were used? More or less what is asked in the title. I am familiar with how the system itself works, but I am wondering if there would be any issues if most chess tournaments didn't follow the Swiss format and to a lesser extent, round-robin and match formats which all have each player play a similar number of games. However, consider a single-or double-elimination tournament bracket, which would mean that when a system was first implemented, stronger players would be playing more games in those tournaments than the weaker players who would be eliminated sooner.

Also the question is not limited to issues in relation to the accuracy of the ratings in a statistical sense, though that would of course be the main consideration.
Theoretically there's no problem. But, for example, if 'parking the bus' approach (a player gets a high level in ranking, then he stops playing) is one of the most known practical issues of Elo ranking, the problem should be with weaker players that play just brackets, with one or few games every tournament. 

Many post-Elo ranking systems, as Glicko (for chess), or TrueSkill (for X-box games), or rankade (our multipurpose ranking system) have some 'activity dynamics feature' to avoid this.

Rankade adds the option for different weights for match, due to different tournament format and/or match duration, and more.

While playing one-on-one (as in a chess group) or in symmetrical factions (as in basketball group), the total number of rating points is conserved. as per Dag Oskar Madsen comment.