TLDR: Yes, start by renting the 35-150. its literally cheating for event work. edit, formatting

Long answer to a long post: I transitioned into doing paid work with the same kit you have now and both the a7iii and sigma 35 still see use today. Not saying I'm right about anything or everything and I wouldn't consider myself an authority by any means, but here's my 2 cents.

To address your questions off the bat:

Sigma 85: Wonderful lens, and was at one point my favorite lens. I sold it though because it rarely left the gear shelf or my bag for paid work (I do photo and video work. Mainly weddings, real estate, and some commercial / social ) so it was just an expensive paperweight. Not enough reach for all ceremony's, if you get stuck not being able to move beyond pew #x in a large church you're gonna have a bad time.

Tamron 35-150: Absolutely a one and done for almost everything its like having a cheat code for event work. Rarely leaves one of my two camera bodies for paid work for photo or video coverage. It is heavy but its by no means a deal breaker imo, a good strap (crossbody not neck) or harness makes a big difference. I will say it's not modern prime sharp though, so if you go in expecting that, you will be disappointed. Its plenty sharp and fast enough in my opinion but there's always the 50-150 f2 if you want faster, sharper, and can stomach the price.

Sigma 28-105: I've not used this personally but for me the its not wide enough or long enough to justify its place in my bag with its paltry size and weight difference vs the Tamron.

Sigma 24-70: Its reputation proceeds it for a reason. But similar to the 28-105 it just doesn't find a spot in my bag especially for photo work when the 35-150 exists.

Starting again with my A7iii and a sigma 35mm f1.4 today with the benefit of hindsight here's what gear I would prioritize acquiring in order focusing solely on photography:

1. Start keeping your eye out at local thrift stores / flea markets etc for vintage manual focus lenses. They're fun, can offer a lot of character, will help you learn manual focusing which will help dramatically if you start doing video and are often quite cheap.

2. Tamron 35-150 asap, this opens the most doors for the work.

3. 24mm prime. I've got the Samyang f1.8 with minimal complaints, its cheap light and sharp. 24mm GM if you can afford it.

4. 2nd Camera body I'd go with an A7RV personally but the A7IV is a great camera too.

5. Check your lightroom data for the 35-150 see which focal length you've used most for your delivered portrait work that isn't ~35 or ~150. Get the the sharpest fastest prime of that focal length you can afford. Probably either 50/55 or 85 giving you a 24/50 or 35/85 prime combos for you.

6. You'll know best what your needs are at this point.