bad idea to charge a Li-ion battery with a lead-acid battery charger
Not really. Both lead acid and Li-ion use a CCCV charging profile. (NiCd and NiMH don't.) The difference between lead acid and Li-ion is just the top voltage. Also, a lead acid charger should be temperature compensated, and a Li-ion charger doesn't need that. Also, many lead acid chargers have two different Constant Voltage stages. While better lead acid chargers end charge if a time-out is reached, better Li-ion chargers end when the current drops below 0.1 C. But, fundamentally, they're generally the same basic charger. Many applications switch from lead acid to Li-ion batteries and keep on using the same charger: forklifts, golf carts, and cell-phone towers.
The main reason we may not use a lead acid charger is not because it's for lead acid, it's because its voltage cannot be adapted for a given Li-ion battery.
do I sacrifice anything by choosing an IC that can charge multiple types of battery chemistry?
No, not really.
e.g. 1000mAh battery should get a 1000mA charging current
Not really. In the absence of a spec sheet, the default charging current should be 0.5 C. E.g., a 1000 mAh cell should get a 500 mA charging current.
Is it better to choose a charger IC based on the battery's maximum voltage or its nominal voltage?
They are two different ways of specifying the same thing. You can see a charger "for 3.6 V cells" or a "4.2 V" charger. They're the same. One charger tells you that its for cells that have a nominal voltage of 3.6 V (implying that the maximum voltage is 4.2 V). The other charger tells you that at 4.2 V it switches to constant voltage mode (implying that it's for 3.6 V nominal voltage cells).
Is one or the other safer and/or better for long-term battery life?
They are both the same. It's only how they're labeled that is different.
a battery whose current capacity
It's not "current capacity". It's just "capacity".
would that be considered wasteful?
No. Nothing is being wasted. You're just getting a longer discharge time before you need to recharge.