Seems that in romantic relationships, desire for connection and continued interest is important to sustaining the relationship or at least the romance.
Yes it is. It helps if you start as friends with several mutual interests. Interest in each other as people goes a long way to staying interested in a relationship. If you can't have a normal conversation with them, you're screwed.
But what's really needed to sustain a romance? Or is that even a worthwhile goal?
You already said whats needed. As per each individual relationship, all that matters is that what you need from the relationship needs to be the same as what your partner wants, or at least mostly. If you're not trying to achieve the same goals, you'll be working against each other.
Its a worthwhile goal if the people in the relationship think it is. If they don't obviously it won't last cause they don't want that.
Do you need constant affection, compliments, honeyed words

, time spent together, or apart to feel connected or loved?
Of course! Not exactly constant though, but we can debate how often constitutes constant some other time.
If you really do love and care about each other though, those things just happen, but it does take some thought - not exactly effort - to remember to keep doing consistently. Though if you find yourself wanting to not give those things, thats another point of trouble.
Can even the simple fact of having an emotional connection be enough?
Nope. If the two of you aren't doing anything to actually make the relationship a relationship (doing things together, talking lots, tending to each others' various needs, etc) then there won't BE a relationship. While its not hard nor trying, it doesn't just happen either.
You're not in a relationship if you just love each other but never do anything. That includes just hanging out.
What is needed to sustain a healthy romantic relationship?
Read above XD. Also, experience helps. I know I couldn't be half the girlfriend I am now if I didn't have A. life skills B. life experience and C. relationship experience. All in all, the fact that this current relationship is my best relationship is due in no small part to having had past relationships. Practice makes perfect!