Should I have succeeded? Machamp solo Attempt

If your lineup was in that order, then my main problem is why did you put your strongest dragonite at the end. It is not going to be taken out, so you are wasting the full potential of it.

Try psycho cut on alakazam so you can dodge, and even zen headbutt with Espeon. You must dodge a few of those HS to survive given that it is harder to dodge (even though doing less damage than a resisted dynamic punch)

And I'm going to be a bit harsh on exeggutor, if it doesn't have legacy confusion it is not worth having on your team. The attack stat is already (in this context) quite low compared to the other options, and the whole point of extrasenaory is to charge up solar beam faster but solar beam is not the winning move in this matchup. I'd swap out the extrasenaory eggy with another Zam or espeon

You should absolutely be dodging, if it'll allow you to fire off another charge move. You'll also need to dodge if all 6 pokemon are at risk of being knocked out. I think that team can do it. Also, use pokebox.

4 Esp/Ala (with FS) + 1 Dnite/Hu + 1 Lugia and then you’ll be fine. No dodging needed.

The time on the phone is probably off by a few seconds. That happens to me when I use "network time" so instead I use the app ClockSync to set it manually. Since then I consistently begin with 176-178 seconds left.
If you can bring more Espeons or Alakazams to the party that would help.
Restart the app before each try (don't know if this is necessary anymore, I had better success previously when I did this but that is some update ago).
Another thing that may help is to see if you can lower the resolution for the game, I use the Game Launcher or Game Tuner for this.
And also, learning to dodge - and when! - is essential.
Against a Counter/Heavy Slam I used 2Alakazam (PC/FS and C/DG), Espeon C/FS, 2Dragonite DT/H, all level 30. I had a Lugia at the last place but it never got any action. I succeeded on my third try with about 12 seconds left, the first two the boss had about 10% HP left. Dodging at the right times made the huge difference.

As others have said, your team looks quite good and more than sufficient on paper. I've been successful with Exeggutor before, but I've had more success without. If the Alakazam is at a comparable level, it will do better. Another thing to try against fighting moves is Gengar he deals neutral damage, but triple resists fighting moves and his attacks are quite powerful even without type advantage.

In general, with espeon/alakazam try to get off 2 futuresights each. This probably means dodging only 1 heavy slam each, but sometimes 2. Heavy slam is the hardest moveset to go against in my opinion. Keep at it! You can do it!

P.S. I'm sorry you don't have any legendaries. Hopefully something changes in the future that you can get them.

You should have 2-3 more glass cannon type attackers.

I use 2 espeons, 1 gengar and Alakazam backed up by an eggy and dragonite.

I'll typically leave gengar last and eggy 2nd to last. As I get closer to the end I start deciding how precise I want to be with dodging. Sometimes I might start eating charge moves if I want to let loose with quick moves and get more energy for a charge. Gengar in particular can tank at the end with extra fighting resistance.