In a certain (deep and amazing) sense, yes, all mathematics does seem to be 'equivalent' to "advanced arithmetic", tho there are some subtleties about which you seem to be confused or unaware.
The details of exactly how any particular mathematical theorem is equivalent to a theorem in arithmetic can be incredibly complex. I don't think your current intuition is doing as much work as you might expect based on your question. Analogously, all chemistry is "basically" physics, tho describing even the simplest chemical reaction in the language (or, rather, one of the languages) of physics would be beyond tedious. It would be similar to describing the history of a country by describing the daily actions of each of its inhabitants or visitors. And in a very real sense, doing so misses the point and does its job poorly and (extremely) inefficiently.
There's also a cultural component that you're missing. 'Mathematics' doesn't (just) cover an ideal Platonic reality consisting of all possible 'mathematic objects'. It is also a human endeavor, thus it consists not only of what we know (or accept, or believe) to be true, 'mathematically', but also of what is suspected or imagined. It encompasses a way of thinking – really many ways of thinking – and it consists of many current and historical philosophies about what exactly (or inexactly) mathematics 'is', essentially or in practice. Further more, it is a process or activity by which people engage in these ways of thinking, argue about the relevant philosophy, and share their thoughts and results with one another.
Altho there are many strong reasons to suspect or believe that human thought is 'fundamentally' equivalent or reducible to computation, currently only (some) humans are capable of participating in and contributing to the ongoing endeavor that is 'mathematics'. And that process, while possibly 'equivalent' to some (Vast) arithmetical calculation, is arguably more richly and productively described exactly as it is now by those in the mathematics community, e.g. talks given, papers published, theorems proved, new fields created (or discovered).
So in many certain (and also deep and amazing) senses, no, mathematics is very much not "basically arithmetic".