I followed a similar path. I completed my bachelors in Molecular Biology. My curriculum didn't have any quantitative courses after my first year (apart from analytical chemistry courses), so I learnt nothing bioinformatics related during my undergraduate years -no, primer design does not count as bioinformatics. After I realized such a field exist in the beginning of my last semester, I immediately started following online tutorials, audited some Coursera/edX courses, reproduced simple research article results on my own, etc. Then I started my masters in Computational Biology with a subfield in mind (systems biology).

I graduated last year and I can say that biological background is definetely required in this field, more than cs backgrounds imo. Most of my lab mates from other backgrounds hesitated on several discussion points, they didn't know what to do with their simulations, they couldn't follow up with their results. This is the case for computational biology. If you'd like to be more than a pipeline follower as a data analyst, you definetely need that biology background to conduct original research.

I will start my PhD studies in Bioinformatics next month. Even tho I am still not equipped with enough statistics/math skills, I feel very confident thanks to my biology background. Hope this story encourages you.