I go Jimmy. If I am just focused on putting a competitive team together that makes the PS and sells tickets (but don't really care about winning it, all that much).
Dame is the more popular player, he has much better durability, and he is a better floor-raiser during the RS. Decent roster-building means you will consistently have decently elite offense too.
However, I pick Jimmy because I think Dame has some serious resilience issues in the PS, and I do not love the idea of him as a #1 option. Lillard has been rather disappointing in the PS compared to his RS exploits.
For example, it doesn't include the 2020 or 21 PS, but for his career Damian Lillard had a -0.8 PIPM for his career. Yes, you read that right, that's a negative alright.
It's not even for his career that he has been an utter disappointment. Dame from 2017-2019 in the PS, scores 24. 2 pts/75 at -0.1 PS adj TS% (below average efficiency). You know who has a better 3 yr scoring peak...Kevin Garnett, had a 3 yr PS peak of 24.5 pts/75 at 0.2 PS adjTS%, and KG gets a lot of flack for not being able to be a go to option on offense in the PS. Lillard's whole game is based around scoring and creation for others, because he is a negative defensively.
Now going back to my point with Lillard, in the RS he has a 6.4 obpm and in the PS a 4.3 obpm, a -2.1 dropoff which is HUGE (2016-19). There is literally not another star in history who comes close to losing as much value. The next biggest dropoff is by David Robinson at -1.5 and then James Harden at -1.4 (before the 2020 PS) who also get a bunch of flack for PS play.
From 2015-2019, Damian Lillard has a -2.4 PIPM... so he is quite literally a negative on the floor. Even his 2019 playoff run where Lillard "took another step," he had a 0.33 PIPM, so still not very good.
From 2016-2020, here is what Lillard looks like from the RS to PS
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Lillard's best attribute is supposed to be his scoring and from 2018-2020 Damian Lillard Playoffs Profile:
23.2
PTS/75 -
5.4 AST/75 -
(-0.1) rTS%
I don't love Jimmy as a #1 option either (although he will surprise me from time to time), however I think his play with Joel Embiid, and Karl Towns gives me faith in him being a 1b or #2 option. He is anywhere from an all-time defender at the SG position in 15 to being around all-defense level in 21. Jimmy is a smart off-ball player, and really gets out in transition, has a low time of possession, cuts well, and just generally finds a way to add value without holding on to the ball.
Jimmy Butler's 3-year peaks in value metrics also just come out much stronger than Dame, because what he does from the RS sticks much better come PS time.
Butler PS Peaks during this span
3-year LEBRON: 4.02 (18-20)
3-year AuPM: 3.7 (18-20)
3-year Backpicks BPM: 5 (18-20)
3-year PS On/Off: 18.8 in 830 MP from 17-19. He also has a 16.5 On/Off from 18-20 in 1397 minutes played.
15-19 PS RAPM: 2.73
Lillard PS Peaks during this span
3-year LEBRON: 0.61 (18-20)
3-year AuPM: 2.8 (19-21) or 1.8 (14-16)
3-year Backpicks BPM: 4.6 (19-21) or 2.8 (14-16/18-20)
3-year PS On/Off: 8.1 (14-16) and (19-21)
15-19 PS RAPM: -1.86
Butler consistently has much stronger on/off in the PS, and with the roles he was tasked with playing during this period, I think it is a good indication he was more impactful. The argument for Dame to me would have to be that if you made him a #2 option, he would immensely benefit from it and add more value than Jimmy, but I am pessimistic he would make such giant leaps.