Wrap Around Scoring for Bowling

In standard ten pin bowling, each game consists of ten frames. In each frame, the bowler can achieve an open frame, a spare, or a strike. An open frame is achieved when the bowler fails to fell all ten pins in two throws, and the score is the number of pins that were felled. A spare is achieved when the bowler takes two throws to fell all ten pins, and the score is ten points plus the number of pins that the bowler fells in their next throw. A strike is achieved when the bowler fells all ten pins in the first throw of the frame, and the score is ten points plus the number of pins that the bowler fells in their next two throws. In order to make the tenth frame equal to the others in possible points, the bowler can make one or two extra throws if they achieve a spare or a strike in that frame; and the pins felled in those extra throws only count as the bonus points for that spare or strike, and not as points in their own right.

As a consequence of this scoring method, there are a couple of side effects that I think are undesirable. The first is that it makes the first frame "special" in that any kind of spare achieved in it is worth exactly the same. For example, if a bowler achieves a spare in all ten frames, there would be ten total first-throws, only nine of which would count as bonus points. Lets say the first throw in the first frame is one pin, and then in every other frame the first throw is nine pins. The bowler's total score would be 190. But if that one-spare occurred instead in any other frame, their score would be 182.

The second problem is that the tenth frame is "special". Besides the obvious point that there can be three throws in the tenth frame, and only two in the others, we have the problem that a failure to fell all of the remaining pins in the third throw of the tenth frame has much less significance than such a failure in any other bowling situation. For example, if a bowler is looking at a 4-7- 10 split in the third throw of their tenth frame, they would be motivated to just fell the 4 and 7 pins and not worry about the 10 pin, because going for the spare would put them at risk of missing completely and the possible reward would be small, just one point. However in any other frame, they would be motivated to try for the spare, because there would be another throw right after it and the reward could be larger, up to eleven points.

Therefore I suggest a simple adjustment. Make the tenth frame the same as all the others, and wrap around the scoring. For example, if a bowler achieves a strike in the tenth frame, let them count the number of pins felled in their first two throws of the game as bonus points. If a bowler achieves a spare in the tenth frame, let them count the number of pins felled in their first throw of the game as bonus points. This solves both of the above mentioned problems, and I think it is more elegant.

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Edit:

The only drawback that occurs to me is that it makes it easier to bowl a perfect game, because you would only need ten strikes in a row instead of twelve. One way of dealing with that would be to have twelve frames, a perfect score would be 360, and you would need to throw twelve strikes to achieve it.