Re: Is this engine big enough for this boat?
Here's a long answer to your fairly short question ...
Too many people for that boat ????? Nah - heck, family and friends out for fun on the boat - if you need space for let's say 9-10 total then that's what you need! leave some at the dock? I wouldn't. You need to get a boat that meets your needs - that's all.
First off, that FW 224 funship can hold 12 I think. heck, 4-5 in the bow EASY and 5-6 in the cockpit - why not? Barely cramped IMO (some would feel overcrowded, but I come from a big family and we pack in well). it has a wrap around bench - plenty of seating. You won't be able to have a dance party onboard with that many, but sounds like an active family+ so it's about crsuising and water sports. Souns like my rew - we load the 19' boat to max allowed capacity all the time and have a great time!
Now to your question. You're on the edge for power for what you plan to use it for. If the design is the same as 2009 model then it has a not-too-deep 17 deg. deadrise which wouks in your favor. Dry weight 4200 # PLUS all gear and people (maybe some of them a pound or two overweight) - that works against you. Duoprop - that will help get you up and going with a load.
You will not know until you test it with a load you'd expect - not so easy to do, I know. So, here's my best guess; you would have not great performance doing the things you want to do. However, I would guess you CAN pull a tube fine with that load - make sure the tube is plenty full of air. Two skiers? Maybe if they are good skiers and on two skiis. New skiiers or slalom skiers - they'll get dragged through the water unless they are good. Smaller load of people, you'd be OK I guess, still not flying out of the hole. It has a wake tower, right? that may help get skiers up if the boat is struggling. Run a lower pitch, four or five blade SS prop for a big gang and water sports and maybe you get it done pretty well.
There are certainly other sports/deck boats out there with similar layouts - but the 224 Deck is a nice one for sure. Getting one with a 5.7L fuel injected (approx. 320 hp) would be the ticket. If you can't and go with the 5.0L injected, you'll have to compromise but you may wind up OK for most things. Still guessing, mind you, with proper set up and folks sitting in the bow to move the weight a bit forward, low pitch 4-5 blade SS prop I'd say you'd be OK and have a ball. Not the fastest, not the best pull for sports, but OK.
Sorry i couldn't give you a specific answer to your question - i doubt you'll get one unless someone has the same boat/engine combo and is maxing it out like you will - maybe they could tell you with more certainty. Or tell us where you live and we'll come test it out with yuo!!!! You might have competition for the purchase though ...... Good luck - don't leave us hanging on what happens because i for one want to know if it works out. Your needs are just like ours and i do love that FW F224 from the boat show!