Solved this, by creating and passing a Udi to the SetValue method. I don't know how I missed this page in the documentation:

https://docs.umbraco.com/umbraco-cms/fundamentals/backoffice/property-editors/built-in-umbraco-property-editors/media-picker-3

Here is the complete code for reference:

public async Task<bool> ResumeSubmission(ResumeViewModel model)
    {
        if (!ModelState.IsValid)
        {
            _logger.LogError("Resume Submission Form Error: Model State Invalid");
            return false;
        }

        var isCaptchaValid = await IsCaptchaValid(model.GoogleToken);

        if (!isCaptchaValid)
        {
            _logger.LogError("Failed Captcha");
            return false;
        }

        try
        {
            string nodeName = $"{RemoveSpecialCharacters(model.FirstName)} {RemoveSpecialCharacters(model.LastName)}";

            var resume = _contentService.Create(nodeName, model.ParentId, "resumeSubmission");

            resume.SetValue("firstName", model.FirstName);
            resume.SetValue("lastName", model.LastName);
            resume.SetValue("email", model.Email);
            resume.SetValue("phone", model.PhoneNumber);
            resume.SetValue("position", model.Position);
            resume.SetValue("message", model.Message);

            if (model.Resume.Length == 0)
            {
                _logger.LogInformation("Resume with no attachment");
                _contentService.Save(resume);

                return true;
            }

            //media file

            var folderId = _mediaService.GetRootMedia().FirstOrDefault(x => x.Name == "Resumes");
            var resumeFile = _mediaService.CreateMedia(model.Resume.FileName, folderId, Constants.Conventions.MediaTypes.File);

            using (Stream s = model.Resume.OpenReadStream())
            {
                resumeFile.SetValue(
                    _mediaFileManager,
                    _mediaUrlGeneratorCollection,
                    _shortStringHelper,
                    _contentTypeBaseServiceProvider,
                    "umbracoFile",
                    model.Resume.FileName,
                    s);

                var operationResult = _mediaService.Save(resumeFile);

                if (operationResult.Success)
                {
                    var udi = Umbraco.Cms.Core.Udi.Create(Constants.UdiEntityType.Media, resumeFile.Key);

                    resume.SetValue("resumeFile", udi.ToString());
                }
                else
                {
                    _logger.LogError(operationResult.Exception, "Error saving media file");
                }
            }

            _contentService.Save(resume);

            return true;
        }
        catch (Exception ex)
        {
            _logger.LogError(ex, "Failure Sending Saving Resume Information");
            return false;
        }
    }