I like the idea of a Test Championship, and there are always going to be issues over teams playing different numbers of tests (and India vs Pakistan never happening). My solution to this is minimum 3-test series, six series per cycle, championship played over 2.5 years, final alternated between countries depending on time of year. It would have the effect of setting the absolute minimum a little higher - just over seven tests per year rather than the current six. More ambitiously, you could increase the number of series per cycle from six to seven or eight. Increasing it to seven would set the minimun average tests per year at 8.4. If you set it at eight, then you can have all teams playing one another every cycle (except India/Pakistan).
This would require:
1. The standardisation of match fees so that all players are paid the same per match, paid for by a central fund to which countries contribute according to means. Something close to this is in the pipeline, which is great. If players know there are at least 8-9 tests per year and they'll be paid, say, £20k to play in each one, that's a proper incentive to want to play test cricket. Financially, it is totally doable if England, Aus and India want it to happen. It also solves the current iniquitous situation where players from a winning team are sometimes paid as little as 10% the amount paid to players from the team they've just beaten. (Central contracts would be unaffected.)
2. England, Australia and India agreeing to play one another home and away over a five-year cycle rather than a four-year cycle. They're not going to want to do this cos money. I actually think twice every four years is a bit too often - twice every five years is about right. But sadly, this is not likely to happen, and without it, a change to a 2.5-year cycle with min 3 tests per series is logistically difficult - you would occasionally have them playing each other twice over one cycle.
Compromises could be made here. You could relax the requirement for three-test series to get the eight series per cycle in, although that would be a shame. England/India/Australia would still be playing each other a lot, given they have five-test series.