Are bikes with aggressive positions required to achieve high fitness levels when training on a bike? Are bikes with aggressive positions required to achieve high fitness levels when training on a bike?

My bike fitter claimed that a more aggressive posture is better for transferring power than a more upright position.

Janheine also agreed that a more aggressive position improves power transfer.
https://janheine.wordpress.com/2018/02/12/myth-5-an-upright-position-is-more-comfortable/

If their claims were true, I would need something like a gravel, road, touring, or cyclocross bike to achieve high levels of fitness.

Interestingly, TrainerRoad doesn't seem to agree.
https://blog.trainerroad.com/does-bike-geometry-affect-power-output-we-asked-an-expert/

Assuming proper fit, gearing, and training zones. If I use a city or hybrid bike, would it really keep me from pushing hard enough for fitness gains?

What if I get a hybrid bike fit instead of a road bike fit?

Imagine if Chris Froome trained on a city or Dutch bike. Could he have achieved a high VO2 max of 88.2?

Speed isn't the only benefit of fitness. Getting fit often improves our health and energy levels.

There are reasons for a more upright position such as comfort, the lack of room for an additional bike, visibility, cheaper brakes levers and shifters, more room for accessories on handlebars, better vision, room for a front basket as the handlebar needs to be higher and of the right shape to accommodate it, and even safely allowing high power output downhill. If your less expensive bike is less aggressive, you might prefer to leave it on a trainer.
TL;DNR - The bike you will ride most often is the bike that will get you fittest. 

Fitness and power transfer are unrelated. Your bike fitter is presuming you want to make the most of the power you have, so is suggesting an aggressive position is needed.  This is possibly because people who pay for bike fitting tend to also be competitive riders who want to go the fastest they can with the power they have. These customers are already close to peak fitness, so they are scratching for low single digit percentages gains in 'power to the ground' (1% faster over an hour's riding is 30 seconds&nbsp;&ndash; that is the difference between winning and not being in the leading peloton). 

If, however, your goal is to get fitter, then a more comfortable position is arguably more suitable. The more comfortable you are, the more likely you are to ride more often and further. If the power you produce is the same on a hybrid at 20km/h as on a fitted racer at 30km, and hour of exercise produces the same results in terms of fitness, it's just that one bike lets you go further. If you are uncomfortable and quit riding the fast bike after an hour vs say an hour and five minutes on the hybrid, the hybrid will get you fitter. 

As far as 'Bike fitness', training on a hybrid for a century on a racer would be unwise: your training would be sub optimal as a more upright position uses muscles differently, but that is a discussion about specific fitness vs general fitness.

How much difference it makes depends on you&nbsp;&ndash; some people find it demoralizing busting a gut to get 20km/h when they know they can do 30km/h on a better bike, and are better off on a faster bike. Others are happy to dial up a work output and accept the speed they get. 

No one needs a 7kg $10K bike to get fit, many think they are fit because they can ride a 7kg $10K bike faster than the guy on the 20kg, $200 BSO. The industry loves selling $10k bikes, hence much of the promotional material around the need for a high end racer.