No one "needs" a Gardevoir to solo Machamp. It's a great Pokemon for sure but Machamp has lots of alternative counters if you don't have a specific Pokemon.
His team is already fine against a fighting moveset Machamp. If there's any issues with winning the raid it's probably going to be against Machamp with Heavy Slam and adding Gardevoir to the team would make it worse.
Psychic is fine. It's obviously worse than Future Sight but there are other things that can impact your damage output such as dodging specials or dodging efficiently, using as few swipes as possible and getting as many attacks in before dodging, and you can work upon those aspects without spending any in-game resources.
If he's doing everything right and still unable to solo Machamp, and moveset is the only thing left that can be improved upon, then consider TMing it.
Do you have a Lugia? She's so good against Machamp. I would start with your 91% Alakazam, then Ho-oh, then FS Espeon and take Lugia. That should be enough if you can't do it with this team. Exeggutor is a net negative and won't get you ahead of the clock. The main point is to get 2 FS off with each Zam/Espeon. Dodging is essential. With Lugia this gets a whole lot easier.
I wish I got Lugia as its one of my favorite legendaries but I wasn't on discord or anything at the time so I missed out on it. Otherwise, I would have powered that bad boy up. The weather in my area is usually clear/sunny so I thought solar beam may be a good idea(Plus gamepress says solar beam is good against machamp).
Your team definitely should be good for close combat machamp. For other moves you will have to get good on dodging with the glass cannons. I would say your goal is to get at least 3 FS total from the Espeon and zam, and 2 SoB from each eggy. One obvious change I would do is tm ZH to confusion,and maybe PC on the zam for better dodging.
Should be fine but you could have difficulty with Heavy Slam as you'll need to dodge and your glass cannons all have quite slow quick moves.
If just one of them faints before getting off a charge move off you might want to consider restarting. Not saying you can't do it, but if it does happen you've got much less room for error afterwards.
Exeggutors are fine, ES/So just about gets it done in time in theory. Doesn't carry the team but it still pulls it's own weight.
Solar Exeggutors are usually seen as something to use when you don't have better counters, but a sunny/clear weather boost will give you a huge increase in damage, they're actually better than Exeggutors with Psychic in this case.
If you're getting really close but still unable to win, consider powering up your Pokemon to their next quick move breakpoints.