If your pain does not improve then have your doctor re-evaluate his standing on the injection. Indeed they can be very helpful, my husband occasionally got them for knee or shoulder pain.
On the other hand I had a friend who used to get them for her elbow pain regularly, both elbow needed surgery later on because of muscle atrophy her nerves had no protection. Your doctor is correct to mention these potential side effects but ultimately the decision is yours.
Here is a quote from an article, and link below to the article.
"Local corticosteroid injection has complications such as infection, sepsis, facial flushing, hypopigmentation, perilymphatic atrophy, bleeding, tendon rupture, steroid flare, soft tissue atrophy, and hypersensitivity reaction."
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC39...
Hypopigmentation and subcutaneous fat, muscle atrophy after local corticosteroid injection (Park et.al.)

I agree with your Doctor
Read this
https://www.medicinenet.com/corticosteroids-ora...

This an afterthought but my Brother in law had leukaemia 3.5 years ago and a bone marrow transplant 2.5 years ago. He was treated with steroids at the time and two weeks ago had a hip replacement on one side but both hip joints are crumbling.

Steroids are horrible drugs

Have you done physical therapy first?? My tendinitis improved from physical therapy

Yes i have it, i wear a brace during the day depending on what im doing. I always do hand and wrist exercises to bulid the strength up. I had a injection in my knee and it didnt help, when i spoke to a gp who specialises in them he said it was very unlikely it would work as it hadn't in my knee. Some peolle they work for and other they don't. Your doctor should allow you to have it they just dont like giving them to you to often. I was 34 at the time and was told i was allowed 2 a year for my knee but haven't had any more as its pointless. Good luck x

I have had shots and it made things worse.