I have always managed with the tools suppplied, I also carry an extensible wheel nut spanner with reversible deep socket 17/19mm, which is very useful for getting the wheel nuts on in alloy wheel recessed holes. At night a torch is essential, so I carry that too. Whenever I buy another car I check that the tools are all present, and that the wheel nuts and the wheels will come off - wheels can rust on and that is best discovered at home, not on a wet night with a puncture. If the car has locking wheel nuts I replace them with ordinary nuts. I ensure that I have a spare wheel that is inflated and road legal.
I last had a puncture in March of this year, around midnight on a wet night. I managed to get the car on to a level spot clear of traffic, and changed the wheel in about 5 minutes. As long as you know what you are doing it should not be a problem. Otherwise it means a long wait until the breakdown services get to you.