Personally I trained capital ship skills on new characters. It is also what people in my alliance recommended when I started training them.
There is little overlap between subcap skills and capital skills so you might as well train it on a separate character. Besides some support skills and very little SP to train a chain of ships to lvl 3 most SP for caps is new: jump drive skills, fighters instead of drones, new armor or shield repair skills for cspital sizes only, etc.
To move a capital ship around you will really want to have a character on a second Omega account to be able to light a cyno in a Recon cruiser.
For me Gerard is my main pilot who can fly all subcapitals well and can light cynos to move my capital ships if needed.
Next, the reason my alliance recommends capital ships on alts is because most capital PvP fights start as subcapital ship fights first, and can escalate or not. You want to be part of the usual action on a subcapital character and have your alt in a capital ship ready if things escalate.
If you only had capital ship skills on your subcapital main you have the choice of either missing all action by being docked in a capital ship in the hopes that the fight today might escalate (which much more often than not doesn’t happen, cap fights are rare!) or you’re stuck in a subcapital ship in the fight as your allies bring capital ships.
Lastly if I ever get to the point of affording and flying supercapital ships on my cap pilot I want to avoid the risk of being stuck in a coffin.
A long time ago supercapital ships could not dock so your character was stuck in the ship logging out and in in space in only that ship. Nowadays supercapitals can dock in Keepstars, but depending on null sec politics these aren’t always available to you and I have seen people in my alliance being stuck in their logged-off supers for some months. I’d rather not have my main stuck and unable to play like that.
To conclude, to fly capitals will want to have a second Omega account anyway to use your ship’s jump drive on your own.
While training capital skills on a good subcap pilot saves a bit of time on support skills you don’t have to train again, many capital skills are new, high SP and specific for caps only so you might as well train it on a new character on the second account.
This has the added benefit that it leaves your subcapital pilot free to do other things like light cynos, fight as the cap waits for escalation or simply play as your supercap pilot is stuck logged off.
I hope this gives you some perspective on your question!
Lastly, I’m having far more fun in subcapital ships and think capital ships are overrated.
I like having caps mainly for their ability to move a whole lot of my ships at once in their Ship Maintenance Bay, and the rare capital fight I’ve had over the years have been a lot of fun, but it’s a big investment of SP and ISK for very little content in my experience. Subcaps are more fun!