AFCI/GFCI is not the most important safety dollar
The ability to be flexible later is. It is making sure you have plenty of spaces in your panel. A 40/42 space panel is the right answer for anything bigger than a cottage, from having seen too many pictures of 30-space panels completely full.
It gives flexibility, and flexibility makes later safety upgrades cheap and easy. I would much rather you fill a 40-space panel with plain breakers (easily upgraded later) than a 24-space with AFCIs (and then you're stuck later, which invites unsafe hacks).
The cost differential (as compared to the panel the electrician probably bid to hit a competitive price) is very low compared to AFCI/GFCI breakers, which I note you are ready to install.
If you have any probability of wanting 2-pole (240V) breakers that are both AFCI and GFCI, avoid GE panels due to a glitch in the way they do AFCI.
You are grandfathered
Grandfathering means that if the circuit was correct at the time it was installed, it does not need to be upgraded merely because the electrical code is revised. The electrician is correct, merely changing the panel is a unit operation and the individual circuits remain "grandfathered". That means yeah, he could actually slap a 16-space panel full of double-stuffs and render you utterly unable to upgrade later.
One reason the electrician may be afraid of AFCI and GFCI on every circuit, is that those devices are sensitive to any wiring flaws - crossed neutral, bootleg ground, you name it. He might install one of those breakers and have it immediately trip. He can't walk away from your house that way (certainly not with payment!) so now he has to go on a "bug hunt" through the house, identifying which loads are on that circuit, opening up every junction box and trying to find the flaw. This could be hours per circuit and will obviously drive the price up considerably. He needs to set your expectations on that.
AFCI/GFCI is still a great idea, though
Applying AFCI to almost every circuit, and GFCI to circuits with a probability of user shock, is still a great way to go, and I want to encourage you to go that way. Just maybe not right away. In my opinion the best way to solve the "bug hunt" problem is to do the work in two phases: First urgently change that nasty Zinsco panel with plain breakers, and then swap them to AFCI/GFCI individually one at a time. (ideally yourself if you can skill up enough to do that, so you can roll it back if problems arise).
To do that, the work done today must make ready for it:
- Enough spaces to accommodate these full-size breakers (not to belabor that point)
- Hot and neutral wires long enough to reach a reasonable variety of locations, so neutrals can reach AFCI or GFCI, and so you have some freedom to relocate breakers. Some electricians, who I call "Captain Snippy", take great pride in nipping back all the wires so they are just long enough to reach their current destinations. He must not do that here.
You could tell him to simply upgrade the circuits to AFCI that give him no trouble, and rollback any troublemakers to plain breakers for now. That will keep costs contained in a way that will make him more comfortable.
He is absolutely correct that refrigerators and freezers should not have AFCI or GFCI protection. They're not the use-case for GFCI, being a grounded all-metal box with inaccessible electricals that you're not likely to drop in a sink. And you don't want to have "dueling safety systems", like a low-oil-level shutoff on a nuclear power plant's emergency generator, or locks on a fire escape so children don't play on it. A GFCI on a fridge just converts the risk from (nonexistent) shock to (likely) food poisoning. Most new fridges are labeled "Don't use with GFCI". AFCI provides a similar risk.
Other installed, hardwired machines like air conditioners also are unlikely to shock you (being well grounded) but likely to nuisance trip.
The only exception is the dryer and range. Many dryers use a 3-prong "NEMA 10" type connection, and ground is bootlegged to neutral inside the dryer. If converting this to NEMA 14 isn't practical (individual ground wires can be retrofitted), then put it on a GFCI breaker, convert the receptacle and dryer cord to NEMA 14 (including removing the bootleg ground jumper), and label the socket "GFCI Protected/No Equipment Ground". Ditto ditto all that for the range.