A few points.
First, as the others on the thread have well-articulated, there are often exceptions to things the Code forbids, particularly for self-defense and defense of the Order writ large, and even outright Code violations still need to be discovered and brought to bear with sufficient political will to punish you. YSMV, but in my sagas the Code is often treated less as "rules to be followed" and more as "reasons to maintain secrecy from outsiders."
Second, there are many adventurous activities outside of what the Code forbids! You have total impunity to go mess with non-Hermetic magical groups. Within the Order, there are also lots of political groups, competitions, or mystery cults to join and/or advance your position within. There are magical locales both ancient and frontier-y to explore and plunder for secrets and cool creatures. For the arguably-standard "young magi found a Spring covenant" saga structure, there are all sorts of adventures to deal with establishing income and resource sources for the covenant, recruiting essential covenfolk and setting up structures to make them self-replacing, and dealing with whatever issues come pre-packaged into your covenant site (regiones, creatures from other Realms who live in or around the covenant, curses or other recurrent phenomena, difficult legal/political considerations with neighbors where dealing with mundanes without violating the Code can be genuinely difficult, etc).
But third, and I think the most important factor that hasn't really been delved into by other commenters and really should be stressed for new players: in Ars Magica, the reclusive parts of the game are fun. Ars Magica isn't like most RPGs, where you've got all the important stuff happening on adventures and the "downtime" mechanics are just to make you better at adventuring. Researching and developing magic in the lab is a fun part of the game, and the primary means by which many (though certainly not all) magi solve problems or pursue their goals. Doing long-term projects, both with and without magic, is (arguably) the core appeal of the game, and certainly the thing that separates Ars Magica from most RPGs.
Ars Magica puts you in the driver's seat of all the inventive capacities that are normally handwaved as the off-screen work of the greatest powers in a setting. It's a lot of fun to decide, say, that you want your covenant to be on a floating island and to actually work out step-by-step what the magical and logistical requirements are both to make that happen at all and then to make it sustainable for the people living there. And these fun-in-their-own-right processes are not mutually exclusive with adventuring, but often a source of adventures in their own right! A difficult magical task might require you to increase your Arts, which motivates you to go on an adventure to find and acquire a high-quality book on the relevant Art(s), and oftentimes these adventures will have costs or consequences that manifest in later adventures down the road, like from enemies you make or favors you end up owing.
So in short, don't write off the reclusive aspect of many magi as "not much of a game," and also don't worry too much because pursuing difficult magical projects will often motivate its own adventures.