More information is required to give useful advice. What is the history of the pain? How did the pain develop; suddenly or gradually? What is your training history? Has there been any dramatic changes in volume, intensity or type of activity? Is there pain at rest or while walking at a slow to moderate pace? What anti-inflammatories were you taking? Oral non-steroidals such as ibuprofen, diclofenac etc? Have you had an ultrasound of the bursa(e), and if so, did it show fibrosis or effusion? What were the exercises and stretches given to you by your physio? Did you do them as suggested?
Modifiable and non-modifiable biomechanical and physiological factors can precede bursitis, or develop due to bursitis. Excessive hip adduction (while walking, opposite side of hip to supporting leg drops down) during mid-stance phase and early gluteus medius/minimus fatigue are two common precipitants and/or consequences of trochanteric bursitis.
Removal of bursae is probably not the next step. Should a biomechanical analysis fail to produce a likely cause, or find a movement pattern which continually aggravates the bursae, then 'relative' rest, +/- corticosteroid injection with a long acting local anaesthetic into the bursae will likely be the next step.
However, a long term solution is still likely to be found in a non-medical approach. We are all a unique biomechanical and physiological machine. We all have our imperfections and asymmetries. The hip pain may be a result of the unique combination of your biomechanics and your chosen activities. This does not mean there is no solution. It may mean there is no one-off, permanent fix. Often a solution is where the athlete accepts that 'Person A' plus 'Activity B' produces 'Pain C' which is adequately self-treated with 'Combination of exercises, stretches, warm-up, self-massage, and appropriate biomechanical alteration to person or equipment D'. It may sound like a simplification, but if your gym sessions lead to a reduction in pain, and stopping the gym sessions leads to the pain returning, then continue the gym sesions.
The greatest cause of failure of non-medical approaches is non-adherence to proposed tretment.
A relative weakness in hip abduction endurance and gluteal inflexibility is often associated with trochanteric bursitis. Pilates instructors are usually pretty good at hammering the hip abductors. Perhaps go along to one basic floor class, and if your abductors are found wanting, continue these exercises at home. They are easy to do on a carpet or rug. Gluteal stretches are fairly simple and person specific. If you have significant gluteal inflexibility, which is more likely in someone with bursitis, then you need to spend more time stretching, particularly where there is the greatest resistance or discomfort during stretching. There is no avoiding this if it is a contributor. Self massage with an old tennis ball can be gold. Initially use just hand pressure on the ball onto the muscle bulk (glut medius and minimus) between the greater trochanter and the (iliac crest) bony ridge on the side of your pelvis. If the tenderness and spasm is sufficiently reduced so that you can lie sideways on the tennis ball, then progress to that. Do the same with glut maximus. Gently stretch before you run and cycle. There is research that suggests that stretching before activity does not decrease the likelihood of injury. Transposing these findings to existing bursitis injuries is flawed. Stretch. 3 months of doing nothing may give you pain relief while you are doing nothing, but all the while you are losing any hip abductor endurance you had, and depending on how much stretching you were doing during this rest period, you will also be losing flexibility which will contribute to pressure/friction at the bursa as soon as you resume activity. Unsurprisingly, the pain will return.
A coach and physio combination will only work if they collaborate. Often egos are the greatest barrier to optimal rehabilitation. Mixed messages will only confuse and create conflict. If you have access to a good physio with personal experience in competitive cycling and/or running, put the challenge to them. Let them know you are desperate and committed to rehabilitation.
It may not be simple, but no less worth doing.