The problem with cherry-picking
Similarly, while an energy saving appliance used in the home might save you money as it may pay for its greater cost through lower electricity bills, it will not reduce carbon emissions. If less electricity is used, the power generation companies will have more allowances, which they will sell to someone else - airlines, maybe, or heavy industry - who will then be able to emit more carbon.
This is the principle of ETS which does encourage lower emissions by reducing the number of allowances over time. In theory, this will encourage industries that buy allowances to become more efficient as the prices of such rises. The scheme is set to avoid price shocks by reducing the numbers of allowances in the system only gradually.
The main problem, there have been smaller ones related to gaming the system as well, with the ETS is that there are too many emissions in the scheme. As a result there is little or no incentive to trade allowances which are below cost for those that need them. There are several reasons for this: firstly, effective lobbying allowed for very generous exemptions and allowances that were initially given to companies (but charged to customers); secondly, post-2008 recession; thirdly, faster than expected buildout of renewable capacity in Germany and elsewhere; fourthly, some but actually very little off-shoring of dirty production to countries outside of the scheme; fifthly, displacement to Poland which has massive allowances due to its traditionally dependence on coal for energy production and inefficient industry. Unsurprisingly, Poland is dead against any early reductions in the number of allowances.
Air travel is a poor example having only recently entered the ETS, something which China, India and the US contest is anti-competitive in itself. It's a moot point as to whether the gradual improvements in efficiency in the airline industry are related to this or just down to very high competitive pressures and the high price of fuel. As a recent entrant to the ETS the air industry has plenty of allowances of its own at the moment so we don't expect to see significant changes as a result of the scheme for a few years yet.