tl;dr: You need to organize/unionize as junior teachers, to deal with this collectively
I would say this is more a question about organizational and power relations than about administrative software. As you've said:
Most of the teaching staff (especially TAs and Tutors) were overlooked ... [even though] ... the majority of student grading that takes place in the university ... is done by this cohort.
and you also mention that this is the case despite individual complaints. You have not mentioned any collective action. Specifically, I assume the following has not happened:
- The junior faculty technical discussion form has deliberated, conducted surveys, and published an official position of the junior faculty regarding administrative software - possibly well in advance of the university having done anything about it, since TAs have been complaining for a while now. This position document lists and explains the problems with "Blackboard", and suggests concrete alternatives.
- The junior faculty liaison to the university's central teaching administration body has pointed out to the administrative official in charge of software selection that the consultation process has not so far included junior faculty/TAs.
- The junior faculty representative on the university teaching oversight committee has brought one of the relevant issues up for formal discussion: Administrative software in general, junior faculty dissatisfaction with the existing software solution ("Blackboard"), or exclusion of junior faculty from the consultative process.
- In negotiations regarding junior faculty employment conditions overall, the issue of suitable administration software has been brought up by the union negotiating team.
None of these happened because either there is no collective entity of the junior faculty (or just the TAs): No union, no intra-university official bodies, no officials (individuals or groups) with recognized official status and actual clout.
You should change that. Perhaps use this opportunity to gather some people disgruntled regarding the administration software issue to commit to some continued concerted activity in trying to form an initial body of the junior faculty, whose first order of business is addressing this issue and approaching management about it. Or rather, the first order of business would be to increase awareness of the issue among other junior faculty, arrange an assembly, have some speakers regarding the need to handle this collectively vis-a-vis university management or the administrative bodies, and so on. It is obviously important to have such an initiative develop into a permanent organizational entity which can go on to address other issues and reach the situation in which some, or all, of the above hypotheticals actually occur. Or better yet, a situation in which your collective presence is felt strongly enough that it would be obvious you must be consulted.
- Even though I said "union", I wasn't talking about strikes, pay, formal labor disputes etc. Unions are not only for those things. At their core they are (or should be) a mechanism for reflection and action on the collective rather than the individual level when this is necessary/useful.
- You might decide (again, collectively) you want to set up an alternative software system in parallel, either for your convenience or for convincing the university to change its mind. This is obviously more likely to be possible when the resources of many people are pooled together.