Is health anxiety something that has developed or gotten worse with age, as your hormone levels dropped, in peri menopause or post menopause. I think a lot of women get less confident and more anxious in general as their hormone levels decrease. Maybe going on HRT will actually make you less anxious.
(edit: Just read Hurdity's comment on your GP's anxiety and totally agree! Maybe she needs to get better informed on the actual risks and benefits of specific types of HRT to alleviate her own anxiety.

)
I've had no bleeding. I guess some women do, but it's not a given. I'd honestly put up with some though in exchange for the benefits I've derived, mostly the boost to my energy levels. I was on my knees before HRT, exhausted, unable to cope....
I'm taking two pumps of Estrogel a day and continuous Utrogestan (though I'm cutting back a bit on that as I've had a few groggy days), which I chose because they are both body identical and have less risks attached. I really have no concerns about health risks!
I've avoided oestrogen in tablet form as they are associated with a risk of stroke and thrombosis. Patches and gels are not as the oestrogen goes straight into the bloodstream. The cancer risk from taking oestrogen would be endometrial, but so long as you're taking progesterone too, you should be covered. I'm going to have a scan once a year to check that the lining of my womb isn't building up just in case. All bases covered. Nothing to lose sleep about IMO!
Utrogestan is the only progesterone available on the NHS. Everything else, whether it's taken alone or in combination with oestrogen, say in a combi patch, is progestin, which is a synthetic form of progesterone. Research indicates that there is a greater risk of breast cancer when taking oestrogen and (synthetic) progestin than oestrogen and progesterone (i.e., Utrogestan). In either case, the risk of breast cancer is minimal though.
https://www.ncbi.nlm.nih.gov/pubmed/17333341 Weighed against the benefits of HRT, especially re protection from osteoporosis and heart attacks, IMO this risk is reasonable and acceptable.
Louise Newson's website (Menopause Doctor) has loads of interesting info on it, and this is her prescribing guide
https://pcwhf.co.uk/wp-content/uploads/2019/03/PCWHF-Easy-HRT-prescribing-guide_March19-update.pdf, which I hadn't seen before I went on HRT but supports the choices I made after doing quite a bit of research. It may not suit you. But it's not a bad place to start.
I do hope you decide to try HRT and aren't put off by your GP. It may not work for you, but IMO it's worth trying. I wouldn't want to live another 20, 30, 40

years putting up with the symptoms you describe and not feeling like me!
S xx