Should I consider something like this
https://skyfiaudio.com/products/sansui-au-6600-integrated-amplifier-super-clean-from-the-70s
I would not. For your Omegas, the suggestion from @danager ,the Music Paradise MP-301 is a better option.
I say this based on your expressed desire for sonic "body and presence". This is an inexpensive integrated tube amplifier that has a consistent word of mouth reputation for good sound quality.
This very simple circuit tube amplifier will pair beautifully (IMHO) with your very simple single driver speaker. I believe the body, presence, tone/timbre and 3-dimensional space will be much improved to what you currently have. All this for less than 500.00 USD. Earlier in this thread I listed some recommendations for you.
All excellent choices for your objectives but more expensive (But worth it). So if you’re leery of spending money then this MP-301 is a very good option to consider. Your jazz will definitely sound better than what you have now.
Charles