In 1989 Fignon of France was in yellow but Lemond (US) beat him in the final contre-la-montre/time-trial and won by eight seconds overall. That was too exciting, perhaps that is why the last stage is a promenade now.
I think you should always try to win, I think it would be good if the last stage was contested.
It's worth pointing out that just because the yellow jersey isn't attacked on that day (although they would be if there was a matter of seconds in it!), there are other jerseys still up for grabs. It's not uncommon for the Green jersey to go down to the final day, for instance in 2005 and 2010. Same goes with the team classification.
The other thing is that it's simply very uncommon for the time gap to be so close on the final stage anyway. In recent years, the 2007 Tour was a good example, as Cadel Evans could have attacked Contador on the final stage. However, Leipheimer (Contador's teammate at Discovery and 8 seconds behind in 3rd place) was faster than Evans, and I suspect that the Lotto team wanted to preserve their second place rather than risking losing everything.
The other aspect is that having the time trial on the penultimate day is also the optimal format. You destroy the riders in the mountains three and two days before, then you let them ride a long transition stage before having the ITT on the penultimate day. We saw how Roglic collapsed in 2020, and the spectacle of the yellow jersey having to chase down a rider in front can be astonishing to watch.
In terms of the mountain classifications, much depends on the place of the climb. There are some climbs that will be categorised as a HC climb regardless of where it falls, but then others might be upgraded if it's at the end of a difficult day.
Speaking of which, I'm always amused by the way that the French translate the names of mountain climbs into French, even when the race is outside of France.
One thing that I'm especially enjoying this year is that both UAE Team Emirates and Jumbo-Visma are really struggling with keeping enough riders in the race. Both of them are down to 6 riders, and Marc Soler (UAE Team Emirates) is now looking in bad shape today. Neither of those teams look capable of controlling the mountains ahead as well!