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Solid tyres are fine for slow moving equipment. High speed tyres wouldn't cope with being solid due to the heating that happens from the distortion and deflection of the sidewalls and the contact patch in coping with the rolling and side forces acting on the tyre.
If you took a car tyre and filled it with anything solid then running it at speed will cause a large amount of heat to build up. To get an idea of how much heat the tyre has to deal with do a long fast run in the car and then immediately stop and feel the rubber. It is quite warm and yet shedding heat as best it can, quite efficiently, being air filled.
The heat comes from the friction of the rubber with the road surface, the flexing of the rubber and from the compression of the air inside. The air is being compressed by the flat spot that travels around the tread of the tyre as the tyre rolls The compression is continuous around the tyre and sends a wave of compressed air around the inside of the tyre. The air movement inside allows the heat generated by the compression wave and from the rubber to be dissipated to the rest of the tyre.
If the tyre was solid then the heat would not be able to dissipate as readily and so the inside of the tyre would oveheat very quickly and then deform. This would lead to instability within the tyre, flat spots and lumps forming and then melting of the tyre core.
In the grand old days of heavy haulage when solid rubber tyres were used the road speed had to be kept very low, 1-5mph depending on load. Over speed those tyres and the solid rubber just melts inside.
The same would happen in a car tyre, just at less load and higher speed.
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