In the 80s and 90s there could be new engines for free practice, then a new one for quali and then a new one for the race. That made reliability worse, whilst increasing costs to go racing, but of course would give a small performance advantage (engine supplier and team dependent).It's not just that though is it? Back in the mid-80s we didn't have all this restricted numbers of engines, gearboxes, tyres etc. I know F1 needs to cut costs but I am sure the cost of completely changing the regs every 5 or so years is massively higher than allowing teams to use a few more engines or tyres in a season... and it would make for better racing. Conserving tyres or engines or gearboxes has always been part of F1, but it's just too much a part of it now for me.
Changing the chassis regs every 5 years or so has little to no bearing on cost as all teams, bar Haas, are spending up to the cost cap limit. And so if all teams have to send money on making a car to the new regulations then it is the same for everyone and so is just one of the costs to go racing. There are very few carry over parts, if any, and each new car is designed and developed for that racing season and so that is the same as designing a car for new regulations. The only real difference is that new regs allow a trade off for developing the current car vs developing the car for the new regs to be made. Which of course benefits the team that is winning and any teams who have a comfortable championship position and can afford to stop developing the current car sooner than other teams.
As an aside, most teams have a larger income than the budget cap and so that results in a bigger pot to pay the drivers, the board and the top 3 salaried employees, amongst other non budget capped spend areas.
I am afraid that I don't have the answer for what you are looking for, as if a few more engines were allowed, teams would keep on pushing the boundaries, have engine failures as they would rather have performance, and then get penalties. Maybe the answer (a bit leftfield) could be a penalty that does not affect the weekend's performance, for example, a reduction in wind tunnel time for the following year?