Should I get a multi-entry visa for visiting Thailand three times a year, or can I manage without it?

That's fine. I do Dec/Jan, April , July , October, Dec/Jan .

I personally wouldn’t rely on that six months a year in Thailand on visa exemptions can be done without any issues for the foreseeable future, so your safest bet is probably to apply for a visa. If you apply for a one year Non O-A visa you just can use income/funds at home to fulfil the financial requirements and therefore don’t need to prove any income or funds in Thailand. A one year Non O-A visa used the right way gives you up to two years in Thailand. There’s a mandatory health insurance, health certificate and police check up requirements.

I’ve been doing this for years.

No problem.

If you qualify for Visa Exempt entry you don't need a Visa for those visits