Time tracking (salaried staff, not hourly)

Hi. I'm asking about time tracking of salaried staff as an assessment of time spent on different projects, services, etc. Wondering how prevelent this is. By this, I don't mean tracking eyes on screens or mouse movement on computers; this is self-entered time with increments of no less than 15 minutes (like a Drucker time-assessment, basically).

BLUF: do managers out there complement MT-style management (O3s, feedback, coaching, delegating, etc) with staff self-reporting time spent, or do you only assess results, outputs, and outcomes? Should I continue without the external requirement for such tracking?

Situation: I have salaried staff working in research support at a university. These staff do not solely serve local researchers, but participate in national service delivery. As such, the university receives a payment for a significant fraction of these salary and benefits from a national organization as defrayment of the costs of our contribution of these "federated" staff to that national service provision. As such, we have been tracking for years the time spent on national work tasks, local work tasks, meetings, etc. in roughly 15-minute increments and reporting that through a regional organization to the national organization, as we were told that was required to justify the financial transfer from the national organization.

Turns out, the national organization was never receiving that, and hadn't required it in that detail at all. No other region nor the other institutions in our region were performing this tracking in the same detail as we were. This tracking was complemented in my local management of these staff (who are individual contributors in the eyes of the university, but may be senior team leaders on national teams) by O3s, feedback, etc (MT).

SO: our local department in which my team resides had a time tracking software package that was ideal for our purposes: easy into which to enter data, easy to generate reports, etc. However, while the whole department formerly reported "positive time" (time worked), they shifted in 2023 to only reporting exceptions (overtime, on-call, illnesses, vacation days). So, they dropped the SaaS subscription to that easy-to-use package. Project teams and my team started using a project management resourcing software that is abysmal as time tracking software; it's designed more for program/project managers to build projects with resources at a less-granular level. The package is not designed or configured for time entry, nor for time reporting. 

Time entry was a small effort of my staff and complemented MT management with a Drucker time assessment, as we thought that was required in order to get co-pay of salaries/benefits. Turns out that was a decade-long lie. Should I continue the "Drucker time tracking" practice, either through building something in a spreadsheet or purchasing the software so recently dumped? I'll likely get push-back on the $2200/yr for repurchasing the time-tracking package, as our top manager dropped it for the department in a cost-cutting effort.

Thanks in advance for any thoughts.