Is it necessary for a Buddhist to believe that existence is suffering?

"It appears that Buddhism depends on a central premise -- that all worldly existence = suffering / dukkha."

This is a correct result of an investigation.

"Are there Buddhist teachers or traditions that teach a joyous approach to worldly life?"

There are many if not most "Buddhist Teacher" and Traditions calling themselves after the Noble Ones. How ever, neither the Buddha, nor his good following desciples, not teacher of the Buddhas Dhamma do such. Why? Becaus they tell what is true and conductive for the path. Every time that one does not see the danger, does not put effort into the training, is wasted time. You can enjoy the reat when work is done.

"Are there any teachings that profess existence is a joyous, if sometimes painful experience, or at least a tolerably pleasant experience? An eternal roller-coaster ride of birth-and-death, joy and sorrow, that one need not seek to escape?"

There are, but not founded in Buddhas Dhamma. Mara constanty teaches such and so those overwhelmed of him as well.

Who ever does not see him/herself the first noble truth, or reject faith, firm faith in it, is incapable to make any long therm conductive and wholesome use of Buddhas teachings, is still a wordling and regarded as outsider.

In regard of the main Question: "Is it necessary for a Buddhist to believe that existence is suffering?" A self called "Buddhist" may believe what ever he wants, and yes, there have always been follower not really knowing the deeper sense of Buddhas teaching and he did not much worry about it.

Speaking from the position of the third (insight) training in Theravada: it is not necessary to believe that existence is suffering; it is important to train insight and see for yourself how things are.

Dukkha (which is perhaps better translated as unsatisfactoriness) is one of the three characteristics of everything (= all mind states -- except of the state of complete extinction of consciousness, nibbana), and is closely related to anatta (non-self, i.e. that you don't have 100% control over your experience) and anicca (everything is changing).

Thus, on a finer level, even positive mind-states (including blissful states of samatha) have a trace of dukkha. However, that does not mean that one should be demolishing everything positive by concentrating on that negative trace. Mindfulness strives to see things as they are, including the positive and negative, without blowing either of them out of proportion. That automatically includes deepening insight into what is dukkha, which is completely different from conceptual conviction about everything being "suffering".

The progress of insight training, as I understand it, is getting gradually beyond the positive/negative/neutral programming (i.e. positive → wanting, negative → avoiding, neutral → switching off; those 3 reactions are traditionally translated as greed/hatred/delusion) and accept with awareness whatever comes as it is, as a part of experience. One can/"should" fully enjoy what is enjoyable, mindfully, without attaching (attachment would cause suffering when the joy goes away), and truly appreciate joy while it is here.

Something like middle way between renunciation and submergence.

The gateway premise is not that "existence is suffering", it is that "something is wrong here". In other words, if you are perfectly satisfied with everything (are you?), if you are clear about the purpose of life, if you are a master in dealing with all possible kinds of situations -- then there is no reason for you to look out for anything at all, be it wisdom or liberation. If that is the case, then you are in a temporary heavenly state (to use Buddhist metaphorical language, wink-wink), which temporarily precludes any motivation to set on a spiritual quest to get to the bottom of things.

However, if you feel that you are not living the life you should be living, if you clearly see society as fundamentally messed-up, and are not willing to be complacent about that, if you are haunted by the underlying feeling of wrongness, of incompleteness, if you feel that something is missing - then obviously you will be very motivated to seek a solution!

Even if life is relatively good - still, if you are any attentive and smart, you may find yourself not being satisfied with superficial success, with solving problems at superficial level -- instead you may want to take a step back and solve the underlying problem, and the problem underlying the underlying problem, and the problem underlying that problem, and so on - you want to get leverage over things, you want to find the fundamental solution, the universal principle... You have a sense that the way you deal with things, however well, still leaves something to be desired - but what?

This feeling of deep buried wrongness, of aching incompleteness, of fundamental restlessness, of subtle unsatisfactoriness - that's what we're talking about here. This is where Buddhism begins: "something is wrong here". This is known as "the ground". You begin with what you have right here and now, the feeling of underlying incompleteness - and you are determined to not stop until you solve this most fundamental of all problems. The having solved the problem is known as "the fruition". And the steps from here to there are called "the path". These are the basic rules of the game. Makes sense, huh? :)