The study of male birth control is publicly available and lists the reasons for the cancellation:
[The decision to cancel] was based on RP2’s review of study AEs [adverse events] and conclusion that the risks to the study participants outweighed the potential benefits to the study participants and to the increased precision of the study outcome findings from having the full cohort contribute to the final analysis.
The AVs of concern to the RP2 were reports of mood changes, depression, pain at the injection site, and increased libido.
The whole list of adverse events and their percentage is listed under "AEs and discontinuation of injections". 4.7% were affected by altered mood, 2.8% by depressed mood or depression, 23.1% by pain at the injection site, and 38.1% by increased libido. Not all of these are definitely related to the birth control.
The question already linked to the study showing a correlation between hormonal contraception and the use of antidepressant. According to the NIH, a side effect of female oral contraceptives can be depression or mood changes (uncommon). Here is a study that showed an increase, but there are also studies that show a decrease in depression with contraceptive use. Here is a review of studies that is inconclusive about a link.
The linked Atlantic article names Liletta (an IUD) as example. According to the manufacturer, depression or depressed mood affect 5.4% of users, 5.2% are affected by mood changes.
A review of studies showed that 22% of users of female contraceptives showed an increase in libido, and 15% showed a decrease. There may be contraceptives which increase libido more, and others which increase it less.
Pain at the injection side is difficult to compare to the various contraceptive options offered to women.
It is true that the study of male birth control was canceled because of observed side effects, but a direct comparison of the severity of side effects of different birth controls is difficult.
There is at least one female birth control product on the market that itself states adverse side effects that are comparable to those of the male birth control in question, at least regarding depression and mood changes. Regarding increased libido, a review of studies has shown that female birth control on average increases libido at half the rate of the male birth control.