Surgeon here, and the short answer is that your GP is doing exactly what the guidance asks of her, not stalling you and not, on what you have described, signalling alarm.
Gynaecomastia can be physiological, driven by medicines or anabolic steroids, linked to a hormonal or medical condition, or simply fatty, and the assessment of any adult presenting with it includes a history, an examination and, where appropriate, blood tests before anything cosmetic is planned. The bloods in a case like yours are typically looking at hormone levels (testosterone, oestradiol, prolactin, and the pituitary hormones that regulate them), along with liver, kidney and thyroid function, because each of those can drive breast tissue in a man. In a 34-year-old with a stable, symmetrical chest of ten years' standing the overwhelming likelihood is that every one of them comes back normal, but the point of the test is to know that rather than assume it. The site's piece on what causes gynaecomastia walks through the list.
The ultrasound does two jobs. It confirms that what is behind the nipple is glandular tissue rather than something else, which matters more when a change is new, one-sided or hard, and it gives a rough sense of the split between gland and fat. That second part is what twelveyearshiding found useful: a surgeon reading a report that says gland both sides already knows liposuction on its own is unlikely to be the right operation. Whether your fullness is gynaecomastia or pseudogynaecomastia is the single question that decides the technique, and imaging is one honest way of answering it.
On going private and skipping it: a responsible private surgeon will want the same workup, and many will not operate without recent bloods on file, because operating on an untreated hormonal cause is a route to recurrence. So the tests are not a detour on the way to surgery, they are part of the road. Three to four weeks for the whole sequence is typical in most systems.
What I cannot tell you from here is what your GP felt when she examined you or why she chose the order she did; that is a conversation to have with her, and it is a perfectly fair question to ask directly at the results appointment.
