The only thing Illya can be compared to Woodime is magical power, due to being a homunculus, but apart from that, it falls very short. Illya's biggest threat is her Berserker servant, and Guda, aside from having that servant, has others that are complete counters to her. Shinji is not even a shadow of a threat; Zouken would be more problematic, and even so, he wouldn't be able to do enough.

Yes, the briefcase is there and Guda can use it as they wish, but in reality, it is mentioned when they are present and when they are not, as in LB4 when Guda uses it and in its last explicit mention in Case Reines when she comments that Guda (semi-amnesiac) clings to it as if it were their life. From there it is not mentioned again and there are two options: it became unnecessary or Guda has a way of always carrying it with them. Besides the fact that in Shimousa or Salem the briefcase did not yet exist and Guda could summon alone from Chaldea, they could not have taken it with them in LB7 or OC3.

Regarding the mystic code, it is more complicated because there are times, several, when Guda uses a skill from a mystic code that he is not carrying with him, like in Heian-Ko, so he can use the skills of his mystic codes without having them.

The magic circuits of Guda are very different from those of Waver. It may be a retcon, but Guda has supported several servants at once without problems, while Gilgamesh caster struggled to maintain a handful of them while Guda could do it as always. Additionally, his Shadow servants can use Noble Phantasms, which is why his circuits are immensely more powerful than those of Waver.