That's what I use grenades for, mainly fire. I'll carry 2 different types in the shield slots or belt, to give more options. Probably because (as others have mentioned) you don't get much for choices of damage type any other way early game.
I always find grenades to be too TU-intensive for my tastes; tricky to get in place and throw in a single turn. Having one pre-primed in your hands at mission start is better, but somewhat risky. The ones that don't need to be primed are certainly more useful, but much more limited in damage type.
With a holstered sidearm, only 5 TU to draw (assuming a free hand) so one can either fire a few shots or shoot once and get behind cover.
That said, you're absolutely right that grenades offer a great wealth of damage types and pack a lot of punch into a single inventory slot. I usually carry both a sidearm and grenades, if possible.
I looked, and all of the anti-tank weaponry (Recoilless Rifles, Rocket Launchers, Panzerfausts, LASSes, HEAT rockets/RPG) are concussive.
R-Rifle is piercing, actually, and (in my opinion) one of the better anti-tanks options. High damage even with normal ammo, 60% armor effectiveness,
and damage directly to armor. All that and you can buy them en masse not too far along, so one can easily stock secondary bases with anti-tank weaponry.
The only real minus is that, unlike the explosive ones, there's no explosion. So only one damage roll, no under-armor shenanigans, etc.
The disposable launchers and basic rocket launcher ammo aren't very reliable at all against proper tanks, no. Thanks to their armor-piercing abilities and blast radius one can have some luck, but not something to be counted on.
Really early on, the Rotogun is a decent option. 133% armor effectiveness is crummy, of course, but it strips armor with every shot... and there are a lot of shots. Even if it fails to take something out, the target ends up a fair bit more vulnerable. Loots UAC chainguns can also serve in this role.
Before getting R-rifles, the light cannon works passably. Not the highest damage, but manageable weight & TU costs, and strips some armor with each shot. Don't use it on an actual tank, but works on lighter vehicles and power armor.
The true best option I've found for dealing with tanks is MAG rockets (custom ammo for the rocket launcher). 240 piercing damage with 60% armor effectiveness can sometimes crack open a tank in a single shot. (The stationary MBTs are another story, of course.) Only real issue (aside from the usual limitations of heavy launchers) is that you need to produce the ammo.
Portable lascannons are also a decent anti-tank option, at least when available.
Re: Shotguns:
I forgot the CAWS, which is nice and some people usually leave enough ammo at your doorstep so you don't have to worry about resupply.
I have never used the scatter ammo, but the AP rounds have a very decent range and damage.
Shoot, I forgot about the CAWS too. The AP ammo makes for a decent rookie weapon, yeah - low skill floor (aim is based on 50+0.5skill instead or raw skill), good damage, ammo is common to find. Only issue is somewhat limited range.