Since my wife has been on her current visa (Tier 2 general) for its
full duration (2 years), she could continue another 3 years and claim
her passport.
No, not exactly. A T2 can apply for Indefinite Leave to Remain (ILR, or permanent residence) after 5 years of bookable time. You need to pad in anywhere from 4 to 6 months for them to consider and approve the settlement application. Once the ILR has been granted, you need to spend a year in that status before nationality can be applied for. Nationality applications are taking between 3 to 6 months to consider and approve, and once it has been approved, you need to book and attend a ceremony which takes anywhere from 2 to 6 weeks, and THEN you can apply for a passport. So on the outside, you're talking about a couple of years. (ILR processing time + 1 year of residence with ILR + nationality processing time + ceremony lead time + passport application processing time).
I have heard of cases where a spouse on a work permit applied for nationality immediately after receiving ILR and did not have to wait the additional year of residency, but haven't seen one myself. In this case you are left with the 4 to 6 month ILR processing time and the 3 - 6 month nationality processing time. In either case all that happens after the first 5 years is the work permit holder can apply for ILR.
So the 5 year requirement is just one of the steps along the path to citizenship.
However, moving to a spouse visa causes this 'tally' to reset...
meaning that she now has to build up her 5 years from year zero (as if
she just arrived in the UK).
They reset the residency clock to zero when a person switches from an economic visa (in your case a Tier 2) to a family formation visa. They created this rule because people could not find T2 work and entered into a 'marriage of convenience' in order to stay in the UK, this was abuse, so they stopped allowing it.
When you take this route, Appendix FM kicks in with the various financial hurdles and the path to ILR is 5 years. However, a spouse can apply for nationality after only 3 years of residence with no requirement to spend a year in ILR status. So you essentially save a year in the path. Everything else I described above with respect to waiting times is applicable.
Are we able to claim an income tax refund?
If you are referring to back taxes paid under a T2, then no. There are no grounds to claim a refund for taxes that fell due on income earned in the UK. This would also be seen as an abusive practice because people would be using their immigration status to escape HMRC. The same is true for the NHS surcharge. These are sunken costs.
