The White Wave group (more often known as Bobo Bandits, due to being from Bobo valley in Xinhe from which the White Wave name comes from, will use Bobo Bandits from here on out) were not part of the 184 uprisings and the ties to the Turbans may be weaker than people realize. I should also clarify the Black Mountain Bandits had no connection at all with the Turbans, they were a very separate group.

The Bobo Bandits were linked to the Turbans in their uprisings in the late 180s but they first appear in 187 as they plundered Taiyuan and Hedong rather than 184. They were a group outside of the official system in Xihe who claimed in the early years some link to the Turbans but by the 190s, were more known as bandits and links may be tenuous to the actual 184 revolts. Rafe De Crespigny in Imperial Warlord: A Biography of Cao Cao notes of their 187 revolt

This seems too far from Zhang Jue's centre of influence in Zhuo commandery to have had any close connection with his teaching or his rebellion; and the group is commonly known simply as the Bobo bandits

The history and motivations of Turban figures and ex-Turban figures (or claimants) and bandits are, alas, not well recorded. None of the Bobo Bandit leaders has their own biographies in the SGZ or the HHS, one of their early leaders Guo Da/Tai disappears after the start of their rise in 187. Until they come into focus via joining the Emperor's army, they only pop up once or twice in the records when it had an impact on other figures of note like allying with Yufuluo or Niu Fu being sent to Hedong against them. Similarly, the Qingzhou Turbans may have become an (unreliable) mainstay in the rise of Cao Cao but we know nothing of their leadership, even a name for any of their leaders has escaped our knowledge.

Was it a case of they had quietly dropped the Turban claim over the years? Was it something they believed in at the start or simply something the leaders in the late 180s sought to tap into to gather support only a few years after the fall of the Zhang brothers but was no longer so useful by the 190s? Were their beliefs similar so somewhat amalgamated into Turbans but not the same as the Turbans of 184 and no reason to be against the Han?

Men like Yang Feng were generally not called Turbans, even when being insulting. Huangfu Li, when using a man of Yang Feng's meagre qualities to shame Li Jue for kidnapping the Emperor used the term bandit as the Lu Bu successfully sought to sway Yang Feng and Han Xian to his side against Yuan Shu by praising their Han service (and also keeping the loot gained from the battle) so he didn't feel they had ideological baggage but that their service to the Han was a way of persuading them away from the recently declared Emperor Yuan Shu

It was certainly not unknown for Turban groups to ally themselves with Han supporting warlords, Huang Shao and Liu Pi's the Runan Turbans allied with Yuan Shu and Liu Bei at various points, Qingzhou Turbans seem to have had a genuine connection to the revolts of 184, a stronghold for the beliefs even if they didn’t rise that year, but after several defeats across the north and Central Plains, they negotiated a surrender and support for Cao Cao, remaining so even after he became controller of the Emperor. The Bobo group had allied with the Xiongnu Shanyu claimant Yufuluo while one former Bobo leader Yang Feng had gone to serve under Han controller Li Jue. Such alliances could bring them a degree of respectability and might have proved useful locally against any rivals.

So really, I wouldn't focus too much on them being Turbans in terms of their attitudes towards the Emperor due to the limited provable connection and the attitudes of those around them.

In terms of why the Bobo group came to the aid of the Emperor, there are some possible reasons though we don't know what Dong Cheng and Yang Feng exactly negotiated with them or King Qubi of the Xiongnu and we don't get anything from the Bobo leaders themselves as their motives. Being an officer at court, a man named a General by the Emperor himself sounds a lot better than Bobo bandit, increasing your local and family prestige, whatever else happens in the future. Yang Feng had been one of them so possible personal connections there, if the Han revived enough to have a stable base then the possibility of being made a Marquis with income (or at least prestige). There was the possibility of power as a protector of the Emperor if they played their cards right, the Han's authority was greatly weakened but it still had some power of legitimacy left.

There isn't a suggestion of the likes of Hu Cai and Han Xian had grand plans for reform, they were local military leaders rather than administrative reformers or ones that seem to have sought to bring in an ideology. Granted, there wasn't much time and resources to do so even after the Emperor's flight, the court was ramshackle, living in ruins and the generals fighting amongst themselves but with their link to the Turbans being tenuous and long since vanished from relevance politically, there isn't anything to suggest they had Turban ideals to reform court. We know so little about the Zhang brothers Turban ideology and plans for the future if they had won while it is hard to know if Yang Feng and co had any inkling of what those would have been either.

In terms of why they didn't seize the Emperor, the Bobo group of Yang Feng, Han Xian, Hu Cai and Li Le were not a united front once they got away from Li Jue and would soon fall into infighting or simply leaving the Emperor's escort. Meanwhile, the Emperor had other commanders under him, Zhang Yang, Dong Cheng, Duan Wei with their armies and resources as well as courtiers. To attempt to seize the Emperor risking making one a target of all the other generals who already didn't trust each other would have been a risky move and made themselves a target for everyone else with an army in the area.