Well, seems you do nevertheless - after all, it was you reaching the gate too late, wasn't it?
It doesn't matter when they announce flights in the lounge - most lounges nowadays - especially those at hubs of an airline - don't announce any flights at all. As a frequent flyer, you should know not to trust any announcements.
Requesting a buggy ride - which is an option to choose, but not a guaranteed benefit (like the car service at the First Class Terminal in FRA for LH F passengers, or the buggy ride in BKK for F pax) - they don't have someone in the lounge monitoring your flight and telling you exactly when to leave - was your own choice. And doing so just 25 minutes before departure is cutting things very close.
Not too close yet if you know the airport well. Assuming gate closure is a fixed 15 minutes before departure, and you have 5 minutes to walk to the gate, that would be reasonable as Plan A, with Plan B being to walk there yourself.
When they announced "in 5 minutes" most definitely it shifted to be their problem. They should simply have told you "Sorry Sir, it might take some time to get your buggy and since your gate is closing in 10 minutes, you should walk there instead". But, you as a frequent flyer should have realised that their "5 minutes" was cutting things too close. The lounge staff isn't the Gate agent. They don't really know whats going on with the flight. There was a good chance of a tight slot, ie a timely departure being highly important, and the pilot would ask the gate to close exactly on T-15min. Even when calling the Gate in this situation, there is absolutely no security the pilot won't overrule.
Because, simply spoken, you were too late.
The blame obviously can be splitted here - and let's be honest, by rebooking you on the next flight, CX admitted they had their fair share of it (unless you were on a fully flex ticket anyway)
So, I hope you learned your lesson. Unless it's a service that is sold as "nanny service to gate" (aka TG F), it's your ultimate responsibility to reach the gate on time.