Like you said the main limit is the production time. 5-7 turns for a modern unit is extremely fast, actually. usually it's more like 10-20 from what I've seen. And it's only 2 techs from bombers to stealth bombers, so your bombers might be obsolete before they can even be built, unless you rush buy them. Ancient units stay current a lot longer.
And realistically, 1 bomber or 1 tank won't do much. They're just a supplement to your main force of infantry. Luckily you can upgrade those to mech infantry, which is a nice unit. But trying to build tanks or bombers is like building a wonder of the world.
That goes more to show how ridiculously cheap World Wonders are in Civ V, not so much that the Modern units are disproportionately expensive.
Five to seven turns would be the normal turnover for a unit like the Paratrooper, which costs 350 hammers. Tanks cost more at 450, and Bombers more still at 520.
Base hammer expectations for a sizable production city in the Modern Era (about size 14 or so) in a suitable location should be about 34-40 hammers (I loaded up saves and counted from the tile yields), without policy adjustments. The production output is slightly higher if you can work SoL Engineers because those give +4 hammers per 2 food.
Windmill gives +15%. Factory gives +50%. Forge gives + 15%. Arsenal gives + 20%. If you went Plastics first into Modern for the Hydro Plant boost, then that's an additional +5 to 7 hammers for ideally situated sites, but we won't count it because it's harder to get those, and you usually have better things to do with riverside cities.
So. All factors considered, building a Tank in the Modern Age is a suitable Military City should base you 35 conservatively, +100% from modifiers for 70 hammers per turn (I have actually observed this), which means that you should be able to compete a Tank in 7 turns.
This is not extremely fast. A situationally better result can be had if you're Persian and in perpetual Golden Age, if you're Iroquoi and have Longhouses, or if you have so many riverside cities that you can afford to just throw one away as a production center for military units only - this site will grant you Hydro Plant for more powah.
I had this Arabia game where I had a city next to hills, a river, and a desert. I built a Hydro Plant, Solar Plant, and Nuclear Plant in it. Maintenance was killer, but it had phenomenal production.
Bombers are harder to build and my experience squares better with you in that regard. With the higher cost and without the benefit of either Forge or Arsenal, the bonus goes down to +65%, and that means that our 35 production only gets boosted up to 55-57 ish, making Bombers take about 9-10 turns to make. That squares with what I see.
Three bombers do a fair bit of damage, though. If you devote three of your cities to making Bombers, you could get a fair number up and running in about 10 turns. If you beelined Stealth, then you could upgrade them as soon as they come online, which is a benefit, but then you're stuck with Infantry and Artillery, which are an era behind.
Instead, you could tech Radar for Bombers, forgo Artillery mostly, and then you can tech Electronics and attack with Mech Inf with Bomber support, which is just fancy.
Attendant production bonuses are on alternate tech lines, so you have to choose whether to have a few high tech units, or more lower tech units.
Also, this scenario is dependent on actually having a production city capable of pulling 70 hammers a turn, just as I described. If you don't have such a city, then you will have problems making Tanks.
