I split my findings into categories and bolded the parts of the studies I found most interesting. I really didn't take the time to be super critical on study design, etc; I was just taking their findings at face value and seeing what, if the study was true as reported, was being claimed. Enjoy!

1. Sleep

https://www.sciencedirect.com/science/article/pii/S1087079216301526

Claim: “Chronic resistance exercise improves all aspects of sleep, with the greatest benefit for sleep quality. These benefits of isolated resistance exercise are attenuated when resistance exercise is combined with aerobic exercise and compared to aerobic exercise alone”

“Improvements in sleep quality were observed across all intensities, however, effect sizes for all sleep quality outcomes (significant and non-significant) tend to be larger in studies of high intensity exercise”
The least benefit was sleep duration
Sleep quality was ‘perceived sleep quality’ - measured by subjective ratings
Chronic here means avg. Of 14 weeks with an average of 60 min per session
“There is limited evidence that combined exercise is better than aerobic exercise alone, and further study is warranted. Data from this review and recent reviews of aerobic exercise [44, 57] suggest that both modalities are effective for improving sleep quality “

2. Cognitive Function

The impact of behavioral interventions on cognitive function in healthy older adults: A systematic review

https://www.sciencedirect.com/science/article/pii/S1568163718301831?via%3Dihub
“Numerous experimental studies have demonstrated that exercise interventions improve cognitive abilities (Kelly et al., 2014b), but benefits have been more consistent for aerobic training, or combined cardiovascular fitness and resistance training”
“Aerobic training (n= 13) most consistently transferred to executive function, while strength/resistance training (n = 8) most consistently transferred to cognitive inhibition and visual working memory. Similarly, aerobic/resistance combination training (n = 6) showed most consistent improvements in visual working memory”

Acute Effects of Resistance Exercise on Cognitive Function in Healthy Adults: A Systematic Review with Multilevel Meta-Analysis.

https://www.ncbi.nlm.nih.gov/pubmed/30838520

“Compared to NEX, RE had a positive effect on global cognition (SMD: 0.56, 95% CI 0.22–0.90, p = 0.004), but was not superior to AE (SMD: − 0.10, 95% CI 0.01 to − 0.20, p = 0.06)”

NEX = no exercise; AE = aerobic exercise, RE = resistance exercise

“Regarding cognitive sub-domains, RE, compared to NEX, improved inhibitory control (SMD: 0.73, 95% CI 0.21–1.26, p = 0.01) and cognitive flexibility (SMD: 0.36, 95% CI 0.17–0.55, p = 0.004)”

Note — the reference to cognitive inhibition was referred to by the study above as well. Kind of interesting.

“Our review detected no difference between RE and AE in acute effects on brain function. This result is in line with the meta-analysis of Northey et al. [15], who investigated the effects of chronic exercise on cognition in adults aged 50 years and older.”

Lifting cognition: a meta-analysis of effects of resistance exercise on cognition

https://www.ncbi.nlm.nih.gov/pubmed/30627769
Positive effects on composite cognitive scores, screening measures of cognitive impairments, and executive functions. no effect on measures of working memory.
“High heterogeneity was observed in all analyses” and the authors weren’t sure why

Long-Term Effects of Resistance Exercise Training on Cognition and Brain Volume in Older Women: Results from a Randomized Controlled Trial.

https://www.ncbi.nlm.nih.gov/pubmed/26581787
Aerobic exercise training has been shown to attenuate cognitive decline and reduce brain atrophy with advancing age. The extent to which resistance exercise training improves cognition and prevents brain atrophy is less known, and few studies include long-term follow-up cognitive and neuroimaging assessments. We report data from a randomized controlled trial of 155 older women, who engaged in 52 weeks of resistance training (either once- or twice-weekly) or balance-and-toning (twice-weekly). Executive functioning and memory were assessed at baseline, 1-year follow-up (i.e., immediately post-intervention), and 2-year follow-up. A subset underwent structural magnetic resonance imaging scans at those time points. At 2-year follow-up, both frequencies of resistance training promoted executive function compared to balance-and-toning (standardized difference [d]=.31-.48). Additionally, twice-weekly resistance training promoted memory (d=.45), reduced cortical white matter atrophy (d=.45), and increased peak muscle power (d=.27) at 2-year follow-up relative to balance-and-toning. These effects were independent of one another. These findings suggest resistance training may have a long-term impact on cognition and white matter volume in older women.

3. Fatigue

Exercise to reduce work-related fatigue among employees: a randomized controlled trial

Note: This is evidence relating to aerobic, not resistance, exercise.
http://www.sjweh.fi/show_abstract.php?abstract_id=3634
“Analyses of covariance (ANCOVA) revealed that, at T1, the EI group reported lower emotional exhaustion and overall fatigue than the WLC group, however, only according to PP analyses. Both according to ITT and PP analyses, EI participants showed higher sleep quality, work ability, and self-reported cognitive functioning at T1 compared to WLC participants. Intervention effects were maintained at T2 and T3.”
“The exercise intervention consisted of 1-hour low- intensity running sessions three times a week for a period of six consecutive weeks.”

Exercise, inflammation, and fatigue in cancer survivors

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4755327/
Might be hard to generalize, but just thought it was interesting to point out another setting where exercise reduced fatigue

Effects of Different Exercise Modalities on Fatigue in Prostate Cancer Patients Undergoing Androgen Deprivation Therapy: A Year-long Randomised Controlled Trial.

https://www.ncbi.nlm.nih.gov/pubmed/28249801
“Different exercise modes have comparable effects on reducing fatigue and enhancing vitality during ADT. Patients with the highest levels of fatigue and lowest vitality had the greatest benefits.”

4. Stuff I didn’t understand but maybe someone else does

The effects of physical activity and exercise on brain-derived neurotrophic factor in healthy humans: A review

https://www.ncbi.nlm.nih.gov/pubmed/23600729
“Evidence from experimental studies suggested that peripheral BDNF concentrations were elevated by acute and chronic aerobic exercise. The majority of the studies suggested that strength training had no influence on peripheral BDNF. The results from most observational studies suggested an inverse relationship between the peripheral BDNF level and habitual physical activity or cardiorespiratory fitness. “
I just included this because I have vague memories of BDNF being related to the psychological effect of exercise, but I'm not well informed enough on this to know the implications.

Takeaways:

There does seem to be some effect of resistance training on cognition. These studies separated cognition into a number of subfields (executive function, visual working memory, etc).
Every time I saw a mention of a study that did a controlled comparison of resistance training+aerobic exercise to resistance training alone, the study concluded that these were similar. It really gave me the sense that when it comes to cognitive benefits and fatigue, high intensity aerobic exercise provides the benefit, and resistance training, if it includes high intensity aerobic exercise, can capture that benefit.
The one cognitive effect that was captured by resistance training but not aerobic exercise was cognitive inhibition, which another study also mentioned. I didn't have time to check how robust a result that was.

One final comment - while I couldn't really make a strong case for resistance training being obviously better than aerobic training with respect to productivity in any manner, I still think that 1) resistance training probably does have absolute benefits to cognition and energy levels, and 2) there are many reasons to pursue resistance training outside of productivity (for instance, things like osteoporosis risk didn't come up in a productivity-related search, but would be relevant when I design my own workout plan).

Exercise (Physical)