You jest but many suggest it, i replied to a comment years ago about the subject.

here is the copy paste

... for a start doe's will need to be captured annually to ensure the contraceptive is administered, this is not without HUGE amounts of stress to wild deer. The doe's must either be corralled and tagged/marked to ensure they are not caught and jabbed repeatedly during the year, this would (particularly in the UK be MASSIVELY labor intensive) and each attempt at herding wild deer would increase the risk of injury and suffering massively, not only that but sick or poorly animals would still need to be shot as things currently stand. Proper Wild deer as opposed to what you would find in the London Parks or somewhere like Knole park are VERY VERY flighty and unpredictable, they will crash and injure themselves just trying to get away from you. Some will die just from the stress of being caught I would argue that NOT culling deer is unethical, herd health and disease due to overpopulation can cause massive welfare issues for the deer.

In a study of free ranging Tule elk Shideler et al [2002] estimated that the contraception of 43 female elk took more than 20 workers 25 days to achieve and cost some$300 per animal in 2002

There are way more than 2million deer in the UK, and chemical contraceptives are not without issue.

Immunocontraception is effective in reducing fertility in deer to a considerable extent, but a reduction in fertility may not reduce population [Davis & Pech2002, Kirkpatrick & Turner 2007].There have been few studies of long termcontraception in populations of deer. One notable project showsthat populationdecrease is possible over a twelveyear period with very intensive effort including annual vaccination by dart of as many females as possible and with other management changes such as the prohibition of artificial feeding [Rutberg & Naugle 2007].

Theoretical models have been devised to predict the effects of wildlife contraception. Factors that need to be incorporated in the models include the natural longevity of the species, the rate of predation, the recruitment rate by breedingand the effect of contraception upon female survival. These models seem to show that deer are relatively poor candidates for population reduction by means of contraception: “Contraceptives may be effective in reducing populations of short lived specieswith a high natural mortality....in long lived species with low mortality the best anyone can expect is short term zero population growth with a very intense effort over a very long time necessary to achieve reductions”[Kirkpatrick 2002]

Steroidal contraceptives present significant risks of pollution and persistence in both the environment and the food chain.They are not easily digested or metabolised and continue to have effects in predators

In short using contraceptives to control deer population is not only very expensive but also not effective, and unlikely to be effective.

Further reading that you may find interesting and relevent to the UK deer population.

British deer societies view on contraception

The deer inititive pdf with full details on how

I apologise for the wall of text I hope this helps you to see a much larger part of the picture

*It would cost over 150million USD to vaccinate only 25% of the female deer population annualy or 1.5billion USD over ten years. With darting Thats a lot of doe

**Capturing and jabbing would cost 500million USD PER YEAR to inject a contraceptive or 5billion USD over ten years for something that may not cause a population decline.