If there's one thing Elio shares with Lusamine, despite his extreme mistrust of this woman that's based solely on his instincts, and later validated over time through a series of queues he's picked up on during their myriad conversations concerning Ultra Space-related phenomena, it's that the PC system is a bane to captured Pokemon. They're meant to be free, not cooped up in a ball and lost to time - hoping their trainer notices them.

Thus, he never captures Pokemon other than the ones already in his party (which would explain why he would never complete the Pokédex if a regional professor asked him to). While having a surplus already [6 in the party, 9 in rotation, 10 under his care, and 6 he's befriended and occasionally worked with - 25 in total, not counting that last category], he's still able to take care of each of them properly due to fully knowing their needs and how do approach each of them. So, it's much easier than it looks, especially when they can generally fend for themselves at a moment's notice.

Most of them have balls of their own while others prefer other methods of concealment, such as hiding in his shadow, camoflaguing as something else or being bound by a piece of equipment that's used in every day situations. From the external perspective - it looks like he doesn't have that many 'mons on him. However, should some members feel homesick, he sends them back to his family's main household where they're cared for properly, or in the case of certain members - allows them to wander for a bit before returning to his side through their unique traveling methods. Overall, Elio has no trouble with the surplus in-questions since most of them either do their own thing or don't require that much care or attention for whatever reason - usually being self-reliant.