FYI, you might have modulation capability, but you might not also have the stability and purity to do a good job.
New DDS based instruments are probably okay, though they typically suffer from spurs. You'd want to check the SFDR (spur-free dynamic range) spec, or better yet, measure it yourself. Carrier stability should be excellent (crystal controlled, give or take the close-in jitter inherent in the DDS process). Depending on how the modulation is performed, it might not be as clean as desired, but should still be okay for radio I would think. (I don't know -- I wouldn't think DDS type generators would be able to do it by analog hardware, not FM at least. I would imagine they usually have an ADC and handle the modulation digitally. So the bandwidth or update rate of that input will matter. You might not get the full ~100kHz bandwidth necessary for a full commercial FM channel including data streams, but I wouldn't think they'd make something so poor that it couldn't handle < 10kHz AM radio, or narrow band FM.)
Older, analog function generators are typically RC oscillators, and suffer from noise and jitter inherent in the circuit. First of all, the tuning circuit is lossy and noisy (there's a resistor!); second, there's a comparator, which must be fast and high gain to get a precise threshold, but high gain and high bandwidth also imply high noise levels, leading to uncertainty both in amplitude (= where the voltage threshold is) and frequency (= when the threshold is crossed). Therefore, jitter and stability aren't usually very good, perhaps on the order of 0.1% of the center frequency. This is pretty troublesome for AM BCB (0.1% of 1MHz is ~1kHz, i.e., expect a loud hiss or rumble behind any modulation you introduce!), and impossible for FM BCB (which has a channel that wide already -- the rumble/hiss will completely dominate!).
The stability of RC oscillators is also typically poor, so you might just manage to set the frequency to what your radio is tuned to (assuming you have an adjustable narrowband sort of receiver -- a wideband e.g. SDR might track a wide swath however, and not mind), but it'll drift over time, due to thermal variation, line voltage, just residual stress in the control pots, etc...
Reference: my primary function generator is a Wavetek 193. It's pretty good (all the functions you need, including sweep/mod, up to 20MHz), but certainly doesn't have the stability needed for radio work. Tuning over its output, you basically just find a wad of hiss. It's also fairly leaky, due to lack of shielding and filtering (it's inside a plastic case) and poor (RF) grounding of the connectors. (In other words, it might be unplugged from everything but AC power, but still be perceptible at the fundamental or harmonics.)
Even a basic LC oscillator has orders of magnitude better stability (I've played with an old Eico 322 RF generator, which has a vacuum tube oscillator, selectable coils for bands, and a variable capacitor for tuning). The AM/FM noise from one is typically imperceptible, and dominated by incidental noise (supply ripple, mechanical resonances -- tap on the enclosure and it goes "BROOongngng"!). Tuning to the fundamental or a harmonic results in strong quieting (i.e., the quiet, stable carrier, unmodulated), just as you should expect.
You should expect the same result (or as good as, or better) with a DDS, at least around the fundamental (since spurs are a thing), but also with pretty good modulation as well (at least within whatever the modulation bandwidth/speed/capability of the instrument is).
Tim